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COORDINATE GEOMETRY

-Is the conic section whose eccentricity is equal to zero.
-Coordinate geometry is the study of representation of geometry figure either on two or three dimension under the one of the following steps.
1. DISTANCE BETWEEN TWO POINTS
-Let’s consider point A EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYon XY- plane now we need to find the distance from A to B.
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Pythagoras theoremEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
But EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Now,
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
For case of three of dimension
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
MID – POINT
– – A midpoint of line segment, is the point bisect a line segment.
OR
– Is the point which divide a certain line into two equal parts.
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

Proof,
-Consider point A(x1,y1), B(x2,y2) and R(x,y)
-Consider the figure lie low.
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
From, similarities of ΔADR and ΔRCB.
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

but,
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
From equationEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
But
AD=X-X1

RC=X2-X
For X
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
For EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
From equation EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
But
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Then,
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EXAMPLE.
1. Find the distance between A EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand B EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
2. Show that the point EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYare vertices of isosceles triangle.
3. Show that the point EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYare vertices of right angled triangles.
4. Show that the point EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand (0,EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY1) are vertices of square.
Solution.
Given
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY


Since EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Then the point A (1,2), B (3,5), C (4,4) are vertices of isosceles triangle.
Hence shown.
TRI – SECTION
-Is the process of dividing a certain line into three section or equal parts
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY



Where the coordinate P intersection depends on the two conditions.
i) When P is close to A
ii) When P is close to B
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
For P close to B.

EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
RATIO THEOREM
– -Is the theory based on a division of a lines segment either internally or externally.
1. INTERNALLY DIVISION
Is a division of a lines segment internally under the given condition of ratio.
-Let line AB being divided at R in ratio M: N.
Where
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Ratio M: N.
-Consider the figure below.
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
From similarities of ΔADR and ΔRCB.
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
II. EXTERNALLY DIVISION
-Is the theory based on a division of a lines segment externally under the given ratio.
Let,
A (X1,Y1), B (X2,Y2) and R (X, Y) under ration R (X,Y).

EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
-Consider the graph below.
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY


EXAMPLES
1. Find the coordinate of point EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYdivided internally or externally in the ration EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
2. Find the point of in-section of a line a joining, point EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYif P is closed of A.
3. Find the coordinate of in – section of a line AB at point P. If B is closer to A given that AEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand BEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
SOLUTIONS
1. For internally division

EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY


EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
GRADIENT
– Is the ration expressed as vertical change over horizontal change.
OR
– Is the ratio between change in Y over change in X.

Mathematically
gradient denoted as EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
i.e EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
However the gradient can explained by using three different methods.
i) . GRADIENT FROM ANGLE OF INCLINATION
ii). GRADIENT FROM THE CURVE (calculus method)
iii). GRADIENT BETWEEN TWO POINTS.
– Consider the figure below.
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
GRADIENT FROM ANGLE OF INCLINATION
-Let θ be angle of inclination
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY


GRADIENT FROM THE CURVE
This is explained by using calculus notation idea where;
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
of a curve at a given point.
However gradient can be obtained directly from the equation of a straight line as coefficient of x from the equation in form of
y =mx + c
BEHAVIOUR OF GRADIENT BETWEEN TWO POINT
-Lets two points be EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYwhich can be used to form the line AB.
(i) If EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY, and EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYthe line increase from left to right imply positive slope.
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY


(ii) If EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY the line decrease from right to left imply EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYslope.

EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY


(iii) If EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYthe line is horizontally with zero gradient
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

(iv) If EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY the line is vertical with infinity gradient
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
COLLINEAR POINTS
– – Are point which lie on the same straight line
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Where,
A, B, and C are collinear.
– – The condition of collinear points have the same slope/ gradient

EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Note:
If A (EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY); B EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand C EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYare collinear then the area of EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY= 0.

Example 1
1. 1.Determine the value of K such that following points are collinear:-
a) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
b) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
2. Show that the points EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY, and EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYare collinear.
3. The straight line EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYCut the curve EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYat P and Q. Calculate the length PQ.
4. If A and B are products of OX and OY respectively. Show that xy=16. If the area of EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYis 8 units square.
Solution:
A EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYBEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY, C EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
For collinear point
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY


2. Give
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY B EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYC EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Alternatively
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY


Since the area of ΔABC is 0 unit hence the points are collinear.

3. Given
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

4. Given
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Since
Area of ΔOAB = 8 square units
Then,
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

5. If EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYare the collinear of midpoint of the line forming the points EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYshow that, x-y+1=0.
Solution
M.P = (x, y)
A EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand B EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Required EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY


EQUATION OF THE LINE
– Mostly depends on different format those under one of the following
1. EQUATION OF A POINT AND SLOPE.
Let point be EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand slope be M.
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
-Let the two points being denoted as AEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY and B EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
-Consider the figure below
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
For the line only if
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY = EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

Multiply by EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYboth side
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY= EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
3. EQUATION OF A SLOPE AND Y – INTERCEPT FORM
– -Consider the slope EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY– intercept, EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY


4. EQUATION OF 2 INTERCEPTS FORM
-Let consider X – intercept C, and Y intercept B
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

5. FAMILY EQUATION
– Is the equation formed from intersection of two lines passing through a certain points.
E.g. equation for intersection of L1 and L2 passing through points (a,b) can be obtained by using the formula.
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Where K is constant
*
Important steps of determining family equation.
1. Solve for K by regarding equations of two lines and the point passing through.
2. Form family equation by using the value of K without regarding the point passing through.

Example.
1. Find the equation passing through the point (2,3) from the intersection to provided
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Solution.
Point EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
But,
(EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
From
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYThe equation is EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
ANGLE BETWEEN TWO LINES
-Let consider EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYwhere Q is angle made between EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYis the angle of EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYis the angle of EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY.
-Consider the figure below
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Our intentions to find the value of EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYwhich always should be acute angle.
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
where

EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY


PARALLEL AND PERPENDICULAR LINES
(a) PARALLEL LINES
-Are the lines which never meet when they are produced
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Means that EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYis parallel to EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYsymbolically EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY//EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
– -However condition for two or more lines to be parallel state that they posses the same gradient.
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

b) PERPENDICULAR LINES
– -Are the lines which intersect at right angle when they are produced.
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Means that EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYis perpendicular to EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
– -Symbolically is denoted as L1L2
– -However the condition for two or more lines to be perpendicular states that “The product of their slopes should be equal to negative one”.
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
– -Let consider the figure below

EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
NOTE:
1. The equation of the line parallel to the line EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY= 0 passing through a certain point is of the form of EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY. Where EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY– is constant.

2. The equation of the line perpendicular to the line EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYpass through a certain point is of the form of EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYwhen EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY– is constant.

3. The calculation of K above done by substitution certain point passing through.
THE EQUATION OF PERPENDICULAR BI SECTOR
– Let two point be A and B.
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Where,
Line L is perpendicular bisector between point A and B.
Now our intention is to find the equation of L.
IMPORTANT STEPS
1. Determine the midpoint between point A and B.
2. Since L and EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYare to each other then find slope of L.

for
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
3. Get equation of L as equation of perpendicular bisector of EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYby using EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand mid point of A and B.
THE COORDINATE OF THE FOOT OF PERPENDICULAR FROM THE POINT
THE POINT TO THE LINE
– -Our intention is to find the coordinate of the foot (x,y) which act as the point if intersection of EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY.
– Let consider the figure below.
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

IMPORTANT STEPS
1. Get slope of formatted line i.e. EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYthen use if to get slope of L2. Since EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
2. From equation of EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYby using EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand point provided from.
3. Get coordinate of the food by solving the equation EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYsimultaneously as the way Y please.
EXAMPLE
1. Find the acute angle 6. between lines
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
2. Find the acute angle between the lines represented by EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
3. find the equation of the line in which such that X – axis bisect the angle between the with line EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
4. find the equation of perpendicular bisector between A EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand B EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
5. Find the coordinate of the foot perpendicular from EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYof the line EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
6. Find the equation of the line parallel to the line 3x – 2y + 7 = 0 and passing through the point EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
7. find the equation if the line perpendicular to the line EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand passing through the point EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
8. Find the equation of perpendicular bisector of AB. where A and B are the point EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYrespectively.
Solution
Given
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Consider
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
From
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Also EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Recall
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY=EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

θ2tan-1 1
Therefore;
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

2)
Solution
Given
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Factorize completely
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
From
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Recall
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Given
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Consider the figure below
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
From
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
The slope is negative then at x –axis y=0
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
4) Solution
Given A EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYB EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
From
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Also
Midpoint = EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
M.p = EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Then
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
The equation is
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Solution
Given EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
From
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
But
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
For the equation
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
The coordinate of the foot is EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
The perpendicular line from point A EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYto the straight line EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYintersect the line at point B. if the perpendicular is extended to C in such a way that AB = EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY. Determine line coordinate of C.
Solution
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Given
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Let
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
From
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Then
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
The coordinate of is EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYsince point B EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Recall
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Since
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Then
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
For x
Compare off
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
For y
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
The coordinate of C is
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
THE SHORTEST/PERPENDICULAR DISTANCE FROM THE POINT TO THE LINE
Introduction:
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
From the figure above it be loved that PC is the only one posses shortest distance rather than the other as perpendicular to the line that is way, our intention is to get shortest / perpendicular distance from the point P to the line.
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Our intention is to find shortest distance from point EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYto the line EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY.
From
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Since
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY


From the trigonometric identity
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Then
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
But
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

Also
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Since the point REcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY is on the line EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Hence;
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Substitute the value of EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYinto 1 above
Then
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

Apply square root both sides
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY= EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY= + EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

Hence
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

At the point EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Note:
The distance from the origin to the line ax +by + c = 0 is given by;
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Example:
1. i) Find the perpendicular distance from point EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYfor the line EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
2. ii) Find the value of K if perpendicular distance from point EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYfor the line EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYis EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYunits.
3. iii) Find the shortest distance from the origin to the line EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
4. If the shortest distance from the point EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYto the line EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYis 3 units. Find the value fm.
Solution
Given: point EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Line EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Recall:
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
The perpendicular distance Is 4 unit

Solution:
Given point EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

Required k
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Then
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
THE EQUATION OF ANGLE BISECTOR BETWEEN TWO LINES
* Consider the figure below.
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Where, PM and PN are perpendicular distance from point P. which are always equal.

EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Since
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Then
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY= EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
NOTE;
i) for the EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYequation take +ve
i ii) for the EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYequation take –ve
THE CONCURRENT LINES
These are the lines which intersect at the same point.
Example:
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

– where EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYare concurrent line.

– However the point of intersection if concurrent line normally calculated under the following steps.
1. Select two equation of straight line which relate to each other from the those equation provided.
2. The get point of inter section of selected equation as usual. Points of intersection into the third equation in such a way that if the result of L.H.S is equal R.H.S imply that these line are currents lines.
Example;
i.Show that the lines EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY, and EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYare current lines
ii. Determine the value of M for which the linesEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY, EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY– 3 = 0 and EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYare current.
iii. Find the equation of bisect of angle formed by the lines represented by pair of the following.
a) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
b) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYand EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Solution:
1)Given
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
By solving since simultaneous equation
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY)
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY=EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

For the first equation take the it be cones
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Then for the EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYequation take cones from
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY= EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY= EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
The equations of base equation of the angle are
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
THE AREA OF TRIANGLE WITH THREE VERTICES
By geometrical method.
Consider the figure below.
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Our intention is to find the area of EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Now,
Area of EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY= area of trapezium ABED area of trapezium ACED
But area of trapezium EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Also consider, Area of trapezium ABED
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Area of trapezium DCEF
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

Area of trapezium
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Then
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
But simplification the formula becomes
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
If ABC has A (x1, y1), B (x2, y2) and C (x3, y3) for immediately calculation of area the following technique should be applied by regarding three vertices of EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYas A (x1, y1), B (x2, y2) and C (x3, y3)
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Area =EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
TERMINOLOGIES OF TRIANGLE
is the line which divide sides of triangle at two equal points
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
If ABC has A (X1, Y1), B (X2, Y2) and C (X3, Y3) for immediately calculation of area the following technique should be applied by regarding three vertices of EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYas A (x1,y1), B (x2, y2) and C (x3, y3)
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Area =EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
TERMINOLOGIES OF TRIANGLE
1.MEDIAN
Median is the line which divide sides of triangle at two equal points
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Where:
AQ, BR and CP are the median of EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Also G is the centered of triangle
CENTROID FORMULA
Consider the figure below with verities A (x1, y1); B(x2, y2) and C (x3, y3)
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Let line BM divide by point G in ration M: N = 2 : 1
Internally
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
The centroid formula is
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY= EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
2.
ALTITUDE OF TRIANGLE
Are the perpendicular drawn Y vertexes to the opposite side of triangles.
3. ORTHO CENTRE
Is the point of intersection of altitude of triangle.
4. CIRCUM CENTRE
Is the point of intersection of perpendicular bisector of the side of triangle.
EXAMPLE.
1. find the area of triangle with vertices A (0,2), B (3,5) and C (-1, 9)
2. Find the centered of the EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYwhere A (1,1); B (3,2) and C (5,4)
3. From triangle ABC, A (2,1) B (6, -9) and C (4,11) find the equation of altitude through A.
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Area = EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Area of EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY= 12 Square units
Solution 2:

EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY= EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
B(X2, Y2)= EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
C(X3, Y3)= EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Recall
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Solution :
Given
AEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY , BEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY CEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Required the equation of latitude consider the figure below
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Since
MAQ.MBC = 1
MAQ = EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
But
MBC = EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
MBC= EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
MBC =6
MBC= EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Y = EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY+ 1
Y = EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY+ 1
Y= EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY+ EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
THE LOCUS.
Locus is a free point which moves on x – y plane under a given condition.
However the locus can be described by using various properties as vertical line (X = EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY), H horizontal line (Y =B) straight line (Y = mx + C) on X – axis, (Y = C), on Y – axis EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYas well as equation of the circle x2 + y2 + 2gx + 2fy + C = 0 or x2 + y2 = r2
Note:
The point of locus is Locus equal distance means equal distance.
EXAMPLE.
1. Find the locus of P (x, y) which is equal distance from A (2, 3) and B (4, 7).
2. A point P moves so that it perpendicular distance from a line 3x + 5y + 4 = 0. Is proper final to square of its distance from point Q (1, 2) if point (2,1) is
one possible position of P, prove that the equation of locus P is given by x2 + y2 – 8x – 14y -3 = 0
3. Show that A (7, -2) and B (2, 6) are all equal distance from the line 3x -4y – 4 = 0.
4. A point Q moves such that its distance from point (5, 3) is equal to twice. Its distance from the line X = 2. Find the equation of locus.
5. Find the locus of the point which moves so that the sum of the square of its distance from point (-2, 0) and (2, 0) is 26 units.
Solution 1:
1. Given
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
AEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
BEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Then consider
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY= EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY= EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY=EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
= EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY= EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Square both sides
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY+EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY=EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY+ EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY²
Expand
x² -4x + 4 +y² -6y +9 = x² – 8x + 16+ y² – 14y + 49
4x + 8y – 52 =0
The locus is straight line
2.
2. Given that 3x + 5y + 4 =0
P EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Q EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Point (2,-1) one possible position of P.
dPL EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
dPl = ICEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY²
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
dPl= EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY————— (i)
Also
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY= EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY= EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY+EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY²
dPl = K EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY= KEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
But EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY=EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY= KEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY= KEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY= 10K
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY= EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
dDL=KEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY= KEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRYEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY= EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY=EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
6x + 10y + 8= EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
6x + 10y +8 = x² – 2x + 1 + y² – 4y + 4
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY² + y² – 8y- 14y -3=0
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY² + y² – 8y – 14y -3 =0
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY² + y² – 8y – 14y -3=0


Solution 4:
QEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
A EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
L: EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY=2
EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
2QA = QL
Solution 5:
P EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY
Points AEcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY& B (2, 0)

EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY




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1 Comment

  • EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - COORDINATE GEOMETRY

    Peter kapinga, December 28, 2023 @ 12:32 pm Reply

    It is so good for the ststudentsudents

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