Question

Find the value of 'k'. If the co-ordinates of the points A(2, -2), B(-4, 2) and
C(-7, k) are collinear.​

Answer 1

If they are collinear they must have same slope(or you can say, they lie on the same line).

Slope of (2,-2) and (-4,2) = slope of (2,-2) and (-7, k)

⇒ (2 - (-2))/(-4 - 2) = (k - (-2))/(-7 - 2)

⇒ - 4/6 = - (k + 2))/9

⇒ - 4(9) = - 6(k + 2)

⇒ 36 = 6k + 12

⇒ 24 = 6k    ⇒ 4 = k

       Method 2:

As they lie on the same line, they must satisfy the equation of line.

 Slope of line = (2 - (-2))/(-4 - 2) = -2/3

Therefore, equation of line is:

⇒ (y - y₁) = m(x - x₁)

⇒ y - (-2) = (-2/3)(x - 2)

⇒ 3y + 6 = - 2x + 4

⇒ 3y + 2x + 2 = 0

    As this defines the whole line, it must satisfy (x , y) = (-7, k)

⇒ 3k + 2(-7) + 2 = 0

⇒ 3k = 14 - 2

⇒ k = 4