JEE Advanced · 2022 · 28 Aug 2022 · Paper 1 · Three Dimensional Geometry

Let

S

be the reflection of a point

Q

with respect to the plane given by

r = - (t + p) \hat{i} + t \hat{j} + (1 + p) \hat{k}

where

t, p

are real parameters and

\hat{i}, \hat{j}, \hat{k}

are the unit vectors along the three positive coordinate axes. If the position vectors of

Q

and

S

are

10 \hat{i} + 15 \hat{j} + 20 \hat{k}

and

α \hat{i} + β \hat{j} + γ \hat{k}

respectively, then which of the following is/are TRUE ?

Answer: (A), (B), (C)

Solution

The plane equation can be rewritten as

r = \hat{k} + t (- \hat{i} + \hat{j}) + p (- \hat{i} + \hat{k})

.

The normal vector is

n = \hat{i} + \hat{j} + \hat{k}

, so the plane equation is

x + y + z = 1

.

For reflection of

Q (10, 15, 20)

S (α, β, γ)

\frac{α - 10}{1} = \frac{β - 15}{1} = \frac{γ - 20}{1} = - 2 \cdot \frac{10 + 15 + 20 - 1}{3} = - \frac{88}{3}

So:

α = 10 - \frac{88}{3} = - \frac{58}{3}, β = 15 - \frac{88}{3} = - \frac{43}{3}, γ = 20 - \frac{88}{3} = - \frac{28}{3}

Checking:

JEE Advanced · 2022 · 28 Aug 2022 · Paper 1 · One or more correct options