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Question
For what values of k are the points A(8, 1) B(3, -2k) and C(k, -5) collinear.
Solution
`"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=11/22`
`Hence , k =2 or k = 11/22`
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