Finding Missing Coordinate When The Given Points Are Collinear

FINDING MISSING COORDINATE WHEN THE GIVEN POINTS ARE COLLINEAR

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When three points are collinear, and a coordinate is missing in one of the points, we can find the missing coordinate using the area of triangle concept. 

That is, if  three points A(x1, y1) B(x2, y2) and C(x3, y3) will be collinear, then the area of triangle ABC  =  0.

Using the above concept, we can find the missing coordinate in the given points. 

Solved Questions

Question 1 :

Find the value of ‘a’ for which the given points are collinear.

(2, 3), (4, a) and (6, –3)

Solution :

Since the given points are collinear, then area of triangle formed by these points  =  0

(2a - 12 + 18) - (12 + 6a - 6)  =  0

(2a + 6) - (6 + 6a)  =  0

2a + 6 - 6 - 6a  =  0

-4a  =  0

a  =  0

Question 2 :

Find the value of ‘a’ for which the given points are collinear.

(a, 2 – 2a), (–a + 1, 2a) and (–4–a, 6–2a)

Solution :

[2a2 + (-a + 1)(6 - 2a)+(-4 - a)(2 - 2a)] - [(-a + 1)(2 - 2a)+2a(-4 - a)+a(6 - 2a)]  =  0

[2a2 - 6a + 2a2 + 6 - 2a - 8 + 8a - 2a + 2a2] - [-2a + 2a2 + 2 - 2a - 8a - 2a2 + 6a - 2a2]  =  0

(6a2 - 2a - 2) - (-2a2 - 6a + 2)  =  0

6a2 + 2a2 - 2a + 6a -2 - 2  =  0

8a2 + 4a - 4  =  0

Dividing the entire equation by 4, we get

2a2 + a - 1  =  0

(2a - 1) (a + 1)  =  0

a  =  1/2 and a  =  -1

Question 3 :

If the three points (3,-1) , (a, 3) and (1,-3) are collinear, find the value of a.

Solution :

Let the given points be A (3,-1) B (a, 3) and C (1,-3)

Slope of AB  =  (y2 - y1)/(x2 - x1)

  =  (3-(-1))/(a - 3)

  =  (3 + 1)/(a - 3)

  =  4/(a-3)  ----(1)

Slope of BC  =  (y2 - y1)/(x2 - x1)

  =  (-3-3)/(1 - a)

  =  -6/(1 - a) ----(2)

Because the given points are collinear, the slopes in (1) and (2) are for the same line. 

So, they are equal. 

(1)  =  (2)

4/(a - 3)  =  -6/(1 - a)

4(1 - a)  =  -6(a - 3)

4 - 4a  =  -6a + 18

6a - 4a  =  18 - 4

2a  =  14

a  =  7

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