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


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



Let A(x1 = 8, y1 = 1), B(x2 = 3, y2 = −2k) and C(x3 = ky3 = −5) be the given points.
The given points are collinear if


$\Rightarrow 8(-2 k+5)+3(-5-1)+k(1+2 k)=0$

$\Rightarrow-16 k+40-18+k+2 k^{2}=0$

$\Rightarrow 2 k^{2}-15 k+22=0$

$\Rightarrow 2 k^{2}-11 k-4 k+22=0$

$\Rightarrow k(2 k-11)-2(2 k-11)=0$

$\Rightarrow(k-2)(2 k-11)=0$

$\Rightarrow k=2$ or $k=\frac{11}{2}$

Hence, $k=2$ or $k=\frac{11}{2}$.


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