The square of the distance of the image of the point (6, 1, 5) in the line x – 1 3 = y 2 = z – 2 4 , from the origin is [2024]

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Solution

Q.
The square of the distance of the image of the point (6, 1, 5) in the line $\frac{x - 1}{3} = \frac{y}{2} = \frac{z - 2}{4}$ , from the origin is _________. [2024]

Ans.
(62)

Let $L : \frac{x - 1}{3} = \frac{y}{2} = \frac{z - 2}{4} = λ$ be the given line.

Any point on L is given by $P (3 λ + 1, 2 λ, 4 λ + 2)$

Direction ratios of L is given by $3 \hat{i} + 2 \hat{j} + 4 \hat{k}$

Now, AP $⊥$ L so we have

$3 (3 λ + 1 - 6) + 2 (2 λ - 1) + 4 (4 λ + 2 - 5) = 0$

$\Rightarrow 9 λ - 15 + 4 λ - 2 + 16 λ - 12 = 0$

$\Rightarrow 29 λ = 29 \Rightarrow λ = 1$

So, P(4, 2, 6) is the mid point of AB

$\Rightarrow 4 = \frac{x + 6}{2}, 2 = \frac{y + 1}{2}, 6 = \frac{z + 5}{2}$

$\Rightarrow$ B(x, y, z) = (2, 3, 7) is the image of A

Required distance = $\sqrt{4 + 9 + 49} = \sqrt{62}$ .

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