Question

Find the value of $\lambda$ for which the points $(6,-1,2),(8,-7,\lambda )$ and $(5,2,4)$ are collinear.

Answer 1

Set B cuts AC in the ratio $l:1$

$x_2=\frac{l{x}_3+x_1}{l+1}$

$8=\frac{l\left(5\right)+1\left(6\right)}{l+1}$

$5l+6=8l+8$

$3l+2=0$

$l=\frac{-2}{3}$

$\therefore$B cuts Ac in the ratio $\frac{-2}{3}:1$

$z_2=\frac{l{z}_3+z_1}{l+1}$

$\lambda =\frac{\left(\frac{-2}{3}\right)\left(4\right)+\left(2\right)}{\frac{-2}{3}+1}$

$=\frac{\frac{-8}{3}+2}{\frac{1}{3}}=\frac{\frac{-2}{3}}{\frac{1}{3}}=-2$

$\therefore \lambda =-2$