Finding Missing Coordinate When the Given Points are Collinear :
Here we are going to see, how to find the missing coordinate when the given points are collinear.
If three points A (x1, y1) B(x2, y2) and C(x3, y3) will be collinear if the area of triangle ABC = 0 .
Question 1 :
In each of the following, find the value of ‘a’ for which the given points are collinear.
(i) (2, 3), (4, a) and (6, –3)
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
(ii) (a, 2 – 2a), (–a + 1, 2a) and (–4–a, 6–2a)
[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
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