PROVE THE GIVEN POINTS ARE COLLINEAR USING THE CONCEPT SLOPE

The points which lie on the same line are known as collinear points.

In case we have three point and we need to prove that the given points are collinear, we may follow the steps given below.

Step 1 :

Choose the points A and B and find the slope.

Step 2 :

Choose the points B and C and find the slope.

Step 3 :

If the slopes found in step 1 and step 2 are equal, then the points A, B and C are collinear.

Formula to find the slope of a line joining the two points (x₁, y₁) and (x₂, y₂) :

m  =  (y₂ - y₁) / (x₂ - x₁)

Example 1 :

Using the concept of slope, prove that each of the following set of points are collinear.

(i) (2 , 3), (3 , -1) and (4 , -5)

(ii) (4 , 1), (-2 , -3) and (-5 , -5)

(iii) (4 , 4), (-2 , 6) and (1 , 5)

Solution :

(i) Let the given points be A (2 , 3)  B (3 , -1) and C (4 , -5)

If the points A, B and C  are collinear then,

Slope of AB  =  Slope of BC

(1)  =  (2)

Hence the given points A, B and C are collinear.

(ii) (4 , 1), (-2 , -3) and (-5 , -5)

Solution :

Let the given points be A(4 , 1) B(-2 , -3) and C(-5 , -5)

If the points A, B and C  are collinear then,

Slope of AB  =  Slope of BC

(1)  =  (2)

Hence the given points are collinear.

(iii) (4 , 4), (-2 , 6) and (1 , 5)

Solution :

Let the given points be A(4 , 4) B(-2 , 6) and C(1 , 5)

If the points A, B and C  are collinear then,

Slope of AB  =  Slope of BC

(1)  =  (2)

So, the given points are collinear.

Example 2 :

If the points (a, 1), (1, 2) and (0, b+1) are collinear, then prove that (1/a) + (1/b) = 1

Solution :

Let the given points be A(a, 1) B(1, 2) and C(0, b+1). Since the given points are collinear, slope of AB is equal to slope of BC.

1/(1 - a)  =  -(b - 1)

1  =  (1 - b)(1 - a)

1  =  1 - a - b + ab

a + b = ab + 1 - 1

a + b = ab

Dividing by ab, we get

(a/ab) + (b/ab)  =  (ab/ab)

1/b + 1/a = 1

(1/a) + (1/b) = 1

Example 3 :

If R (5, 6) is the midpoint of the line segment AB joining the points A(6, 5) and B(4, y) then y equals

(a) 5            (b) 7        (c) 12       (d) 6

Solution :

R is the midpoint of AB.

Slope of AR and slope of BR are equal.

Slope of AR = (5 - 6) / (6 - 5)

= -1/1

= -1

Slope of BR = (y - 6) / (4 - 5)

= (y - 6) / (-1)

= -1(y - 6)

Slope of AR = Slope of BR

-1 = -1(y - 6)

y - 6 = 1

y = 1 + 6

y = 7

So, the value of y is 7, option b is correct.

Example 4 :

Check whether points are collinear or not A(1, -1), B(5, 2) and C(9, 5).

Solution :

Slope of AB = (2 + 1) / (5 - 1)

= 3/4

Slope of BC = (5 - 2)/(9 - 5)

= 3/4

Since the slopes are equal. Then the points are collinear.

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