Let $S$ be the mirror image of the point $Q(1,3,4)$ with respect to the plane $2 x-y+z+3=0$ and let $\mathrm{R}(3,5, \gamma)$ be a point of this plane. Then the square of the length of the line segment SR is

JEE Main · 2021 · 27 Aug 2021 · Shift 2 · Three Dimensional Geometry

Let

S

be the mirror image of the point

Q (1, 3, 4)

with respect to the plane

2 x - y + z + 3 = 0

and let

R (3, 5, γ)

be a point of this plane. Then the square of the length of the line segment SR is

Answer: 72

Solution

For the plane

2 x - y + z + 3 = 0

the normal is

(2, - 1, 1)

with

∣ n ∣^{2} = 6

. Evaluating the plane expression at

Q (1, 3, 4)

gives

2 - 3 + 4 + 3 = 6

. The mirror image is

S = Q - \frac{2 ( 6 )}{6} (2, - 1, 1) = (1, 3, 4) - 2 (2, - 1, 1) = (- 3, 5, 2) .

Since

R (3, 5, γ)

lies on the plane,

2 (3) - 5 + γ + 3 = 0

gives

γ = - 4

, so

R = (3, 5, - 4)

. Then

S R^{2} = (- 3 - 3)^{2} + (5 - 5)^{2} + (2 + 4)^{2} = 36 + 0 + 36 = 72.

The square of the length of

S R

72

JEE Main · 2021 · 27 Aug 2021 · Shift 2 · Integer answer

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