Question

for what value of K are the points (8,1), (3,-2k) and (k, -5) are collinear

Answer 1

Let (x₁,y₁)=(8,1), (x₂,y₂)=(3,-2k) and (x₃,y₃)=(k,-5) are the given three points. Now if these are collinear then 
|x₁(y₂-y₃)+x₂(y₃-y₁)+x₃(y₁-y₂)|=0
or, |8{-2k-(-5)}+3(-5-1)+k{1-(-2k)}|=0
or, |8(-2k+5)+3(-6)+k(1+2k)|=0
or, -16k+40-18+k+2k²=0
or, 2k²-15k+22=0
or, 2k²-11k-4k+22=0
or, k(2k-11)-2(2k-11)=0
or, (2k-11)(k-2)=0
either, 2k-11=0
or, k=11/2
or, k-2=0
or, k=2
∴, k=2,11/2

Answer 2

Hey,

The answer for this question is 11/2 and 2.

Please do refer the attachment for the steps.