The length of the perpendicular from point ( 1 , 2 , 3 ) to the line x - 6 3 = y - 7 2 = z - 7 - 2 is:

Question

The length of the perpendicular from point $(1, 2, 3)$ to the line $\frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{- 2}$ is:

Smart Achievers · Accepted Answer

(3)

Ans. 7

Explanation:

Let a point on a given line be $A (3 λ + 6, 2 λ + 7, - 2 λ + 7) .$

Let the point $B$ be $(1, 2, 3)$ .

$∴$ Direction ratios of AB $= (3 λ + 6 - 1, 2 λ + 7 - 2, - 2 λ + 7 - 3)$

$= (3 λ + 5, 2 λ + 5, - 2 λ + 4)$

Given, direction ratios of the line are $3, 2, - 2$ .

$∵$ Lines are perpendicular.

$∴$ $3 (3 λ + 5) + 2 (2 λ + 5) - 2 (- 2 λ + 4) = 0$

$\Rightarrow$ $9 λ + 15 + 4 λ + 10 + 4 λ - 8 = 0$

$\Rightarrow$ $17 λ + 17 = 0$

$\Rightarrow$ $λ = - 1$

$∴$ $Point A is (3, 5, 9) .$

$∴$ The length of perpendicular $= \sqrt{{(3 - 1)}^{2} + {(5 - 2)}^{2} + {(9 - 3)}^{2}}$

$= \sqrt{4 + 9 + 36}$

$= \sqrt{49}$

$= 7$

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