Question

If the foot of the perpendicular drawn from the point $(1,0,3)$ on a line passing through $(\alpha ,7,1)$ is $(\frac{5}{3},\frac{7}{3},\frac{17}{3})$, then $\alpha$ is equal to

Answer 1

Given points $P(1,0,3)$ and $Q(\frac{5}{3},\frac{7}{3},\frac{17}{3})$

Direction ratios of $PQ:(\frac{2}{3},\frac{7}{3},\frac{8}{3})$
Direction ratios of line $L:(\alpha -\frac{5}{3},7-\frac{7}{3},1-\frac{17}{3})$
$=(\frac{3\alpha -5}{3},\frac{14}{3},-\frac{14}{3})$

As line $L$ is perpendicular to $PQ$,
so, $\left(\frac{2}{3}\right)\left(\frac{3\alpha -5}{3}\right)+\left(\frac{7}{3}\right)\left(\frac{14}{3}\right)+\left(\frac{8}{3}\right)(-\frac{14}{3})=0$

$\Rightarrow 6\alpha -10+98-112=0$
$\Rightarrow 6\alpha =24\Rightarrow \alpha =4$