MATHEMATICS As LEVEL(FORM FIVE) NOTES – COORDINATE GEOMETRY

COORDINATE GEOMETRY

– Is the conic section whose eccentricity is equal to zero.

– Coordinate geometry is the study of representation of geometric figures either in two or three dimensions under one of the following steps.

1. DISTANCE BETWEEN TWO POINTS

Consider point A and B on the XY-plane. We need to find the distance from A to B.

Pythagoras theorem:

But and and

Now,

For the case of three dimensions:

MID-POINT

A midpoint of a line segment is the point that bisects the line segment.

OR

It is the point which divides a certain line into two equal parts.

Proof:

Consider points A(x₁, y₁), B(x₂, y₂) and R(x, y).

Consider the figure below:

From similarities of ΔADR and ΔRCB:

But,

From equation:

But

AD = x – x₁

RC = x₂ – x

For X:

For Y:

Then,

EXAMPLES

1. Find the distance between A and B .
2. Show that the points , , and are vertices of an isosceles triangle.
3. Show that the points , , and are vertices of right-angled triangles.
4. Show that the points , , and (0, 1) are vertices of a square.

Solution

Given:

Since and

Then the points A (1, 2), B (3, 5), C (4, 4) are vertices of an isosceles triangle.

Hence shown.

TRI-SECTION

Is the process of dividing a certain line into three sections or equal parts.

Where the coordinate P intersection depends on the two conditions:

• i) When P is close to A
• ii) When P is close to B

RATIO THEOREM

Is the theory based on a division of a line segment either internally or externally.

1. INTERNAL DIVISION

Is a division of a line segment internally under the given condition of ratio.

Let line AB be divided at R in ratio M: N.

Ratio M: N.

2. EXTERNAL DIVISION

Is the theory based on a division of a line segment externally under the given ratio.

Let A (X₁, Y₁), B (X₂, Y₂) and R (X, Y) under ratio R (X,Y).

Consider the graph below.

EXAMPLES

1. Find the coordinate of point and divided internally or externally in the ratio .
2. Find the point of intersection of a line joining points and if P is close to A.
3. Find the coordinate of intersection of a line AB at point P if B is closer to A given that A and B .

SOLUTIONS

1. For internal division

GRADIENT

Is the ratio expressed as vertical change over horizontal change.

OR

Is the ratio between change in Y over change in X.

Mathematically, gradient denoted as i.e

However, the gradient can be explained by using three different methods:

1. Gradient from angle of inclination
2. Gradient from the curve (calculus method)
3. Gradient between two points

Consider the figure below:

GRADIENT FROM ANGLE OF INCLINATION

Let θ be the angle of inclination.

GRADIENT FROM THE CURVE

This is explained by using calculus notation where;

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 POINTS

Let two points be which can be used to form the line AB.

(i) If , and the line increases from left to right implying positive slope.

(ii) If the line decreases from right to left implying negative slope.

(iii) If the line is horizontal with zero gradient.

(iv) If the line is vertical with infinite gradient.

COLLINEAR POINTS

Are points which lie on the same straight line.

Where A, B, and C are collinear.

The condition for collinear points is that they have the same slope/gradient.

Note:

If A (x₁, y₁), B (x₂, y₂) and C (x₃, y₃) are collinear then the area of triangle ABC = 0.

Example 1

1. Determine the value of K such that the following points are collinear:
   • a) , , and
   • b) , , and
2. Show that the points , , and are collinear.
3. The straight line cuts the curve at 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 is 8 units square.

Solution:

A B C are collinear points.

For collinear points:

Equation of the Line

Mostly depends on different formats under one of the following:

1. Equation of a Point and Slope

Let point be and slope be M.

Let the two points be denoted as A and B .

Consider the figure below:

For the line only if

=

Multiply both sides by .

3. Equation of Slope and Y-Intercept Form

Consider the slope and y-intercept .

4. Equation of 2 Intercepts Form

Consider X-intercept C, and Y-intercept B.

FAMILY EQUATION

Is the equation formed from intersection of two lines passing through certain points.

Example: Equation for intersection of L₁ and L₂ passing through points (a,b) can be obtained by using the formula:

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

Find the equation passing through the point (2,3) from the intersection provided.

Solution

Point .

But,

PARALLEL AND PERPENDICULAR LINES

(a) PARALLEL LINES

Are lines which never meet when they are extended.

Means that is parallel to symbolically //

However, the condition for two or more lines to be parallel states that they possess the same gradient.

(b) PERPENDICULAR LINES

Are lines which intersect at right angles when extended.

Means that is perpendicular to .

Symbolically is denoted as L₁ ⊥ L₂.

However, the condition for two or more lines to be perpendicular states that “The product of their slopes should be equal to -1”.

Consider the figure below:

NOTE:

1. The equation of the line parallel to the line passing through a certain point is of the form . Where is constant.
2. The equation of the line perpendicular to the line passing through a certain point is of the form when is constant.
3. The calculation of K above is done by substitution of certain point passing through.

THE EQUATION OF PERPENDICULAR BISECTOR

Let two points be A and B.

Line L is perpendicular bisector between points A and B.

Now our intention is to find the equation of L.

IMPORTANT STEPS

1. Determine the midpoint between points A and B.
2. Since L and are perpendicular to each other then find slope of L.
3. Get equation of L as equation of perpendicular bisector of by using slope and midpoint of A and B.

THE COORDINATE OF THE FOOT OF PERPENDICULAR FROM THE POINT TO THE LINE

Our intention is to find the coordinate of the foot (x,y) which acts as the point of intersection of and .

Consider the figure below.

IMPORTANT STEPS

1. Get slope of formatted line i.e. and then use it to get slope of L₂. Since .
2. From equation of by using and point provided.
3. Get coordinate of the foot by solving the equations and simultaneously.

EXAMPLE

1. Find the acute angle θ between lines and .
2. Find the acute angle between the lines represented by .
3. Find the equation of the line such that the X-axis bisects the angle between the lines .
4. Find the equation of perpendicular bisector between A and B .
5. Find the coordinate of the foot of perpendicular from the line .
6. Find the equation of the line parallel to the line 3x – 2y + 7 = 0 and passing through the point .
7. Find the equation of the line perpendicular to the line and passing through the point .
8. Find the equation of perpendicular bisector of AB where A and B are the points and respectively.

Solution

Given:

Consider:

THE SHORTEST/PERPENDICULAR DISTANCE FROM THE POINT TO THE LINE

Introduction:

From the figure above, PC is the only one that possesses the shortest distance as it is perpendicular to the line. Our intention is to get the shortest/perpendicular distance from point P to the line.

Our intention is to find the shortest distance from point to the line .

From

Since

Then

THE EQUATION OF ANGLE BISECTOR BETWEEN TWO LINES

Consider the figure below.

Where PM and PN are perpendicular distances from point P, which are always equal.

Since

Then

THE AREA OF TRIANGLE WITH THREE VERTICES

By geometrical method.

Consider the figure below.

Our intention is to find the area of .

Now,

Area of = area of trapezium ABED – area of trapezium ACED.

But area of trapezium ABED and area of trapezium DCEF are:

Then

But simplification the formula becomes:

TERMINOLOGIES OF TRIANGLE

Median is the line which divides sides of triangle at two equal points.

Where AQ, BR and CP are the medians of the triangle. Also G is the centroid of the triangle.

CENTROID FORMULA

Consider the figure below with vertices A (x₁, y₁), B (x₂, y₂) and C (x₃, y₃).

Let line BM be divided by point G in ratio M: N = 2:1 internally.

The centroid formula is:

ALTITUDE OF TRIANGLE

Are the perpendiculars drawn from vertices to the opposite side of the triangle.

ORTHO CENTRE

Is the point of intersection of altitudes of a triangle.

CIRCUM CENTRE

Is the point of intersection of perpendicular bisectors of the sides of a triangle.