If the image of the point P ( 1 , 2 , a ) in the line x - 6 3 = y - 7 2 = 7 - z 2 is Q ( 5 , b , c ) , then a 2 + b 2 + c 2 is equal to [2026]

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Solution

Q.
If the image of the point $P (1, 2, a)$ in the line $\frac{x - 6}{3} = \frac{y - 7}{2} = \frac{7 - z}{2}$ is $Q (5, b, c)$ , then $a^{2} + b^{2} + c^{2}$ is equal to [2026]

1 298

2 264

3 283

4 293

Ans.
(1)

$Point M \equiv (3, \frac{b}{2} + 1, \frac{c + a}{2}) satisfies the line$

$\frac{3 - 6}{3} = \frac{\frac{b}{2} + 1 - 7}{2} = \frac{\frac{c + a}{2} - 7}{- 2}$

$- 1 = \frac{b - 12}{4} = \frac{c + a - 14}{- 4}$

$\Rightarrow b = 8 . . . (1)$ & $c + a = 18 . . . (2)$

$Now P Q ⊥ L$

$\Rightarrow (4 i + (b - 2) j + (c - a) k) \cdot (3 i + 2 j - 2 k) = 0$

$\Rightarrow 12 + 2 (b - 2) - 2 (c - a) = 0$

$\Rightarrow 6 + (b - 2) - (c - a) = 0$

$\Rightarrow b - c + a + 4 = 0$

$\Rightarrow 8 - c + a + 4 = 0$

$\Rightarrow c + a = 12 . . . (3)$

$From (2) & (3)$

$c = 15, a = 3$

$So a^{2} + b^{2} + c^{2} = 9 + 64 + 225 = 298$

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