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

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Solution

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

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

As line $L$ is perpendicular to $P Q$ ,
so, $(\frac{2}{3}) (\frac{3 α - 5}{3}) + (\frac{7}{3}) (\frac{14}{3}) + (\frac{8}{3}) (- \frac{14}{3}) = 0$

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

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