In each of the following find the value of 'k', for which the points are collinear.\left(\right. 8 , 1 \left.\right) , \left(\right. k , - 4 \left.\right) , \left(\right. 2 , 5 \left.\right)

Foundation

Mathematics Foundation

distance formulae

Coordinate Geometry

Question

In each of the following find the value of 'k', for which the points are collinear.

(8, 1), (k, - 4), (2, 5)

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L e t

A (8, 1) = (x_{1}, y_{1})

B (k, - 4) = (x_{2}, y_{2})

C (2, - 5) = (x_{3}, y_{3})

A r e a o f T r i a n g l e

A B C = 0

∴ A B C

i s a s t r a i g h t l i n e .

= \frac{1}{2} [x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2})]

= \frac{1}{2} [8 (- 4 + 5) + k (- 5 - 1) + 2 (1 + 4)] = 0

= \frac{1}{2} [8 (1) + k (- 6) + 2 (5)] = 0

= \frac{1}{2} [8 - 6 k + 10] = 0

, \frac{1}{2} (18 - 6 k) = 0

9 - 3 k = 0

- 3 k = - 9

3 k = 9

∴ k = \frac{9}{3} = 3