Question

For what values of k are the points A(8, 1) B(3, -2k) and C(k, -5) collinear.

Answer 1

`"Let" A(x_1=8, y_1=1) , B (x_2=3,y_2=-2k) and C (x_3=k, y_3 = -5) `be the given points 

The given points are collinear if

`x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1+y_2)=0`

`⇒8(-2k+5)+3(-5-1)+k(1+2k)=0`

`⇒-16k+40-18+k+2k^2=0`

`⇒ 2k^2 -15k+22=0`

`⇒ 2k^2 -11k -4k +22=0`

`⇒ k (2k-11)-2(2k-11)=0`

`⇒ (k-2)(2k-11)=0`

`⇒ k=2 or k=11/2`

Hence, `k =2 or k = 11/2`