If the mirror image of the point $(2,4,7)$ in the plane $3x-y+4z=2$ is $(a, b, c)$, then $2a+b+2c$ is equal to :

JEE Main · 2022 · 29 Jun 2022 · Shift 1 · Three Dimensional Geometry

If the mirror image of the point

(2, 4, 7)

in the plane

3 x - y + 4 z = 2

(a, b, c)

, then

2 a + b + 2 c

is equal to :

(A)54
(B)50
(C)-6✓ Correct
(D)-42

Answer: (C)

Solution

The mirror image of point

(x_{0}, y_{0}, z_{0})

in plane

A x + B y + C z = D

is found using:

\frac{x - x _{0}}{A} = \frac{y - y _{0}}{B} = \frac{z - z _{0}}{C} = \frac{- 2 ( A x _{0} + B y _{0} + C z _{0} - D )}{A ^{2} + B ^{2} + C ^{2}}

For point

(2, 4, 7)

and plane

3 x - y + 4 z = 2

\frac{a - 2}{3} = \frac{b - 4}{- 1} = \frac{c - 7}{4} = \frac{- 2 ( 6 - 4 + 28 - 2 )}{9 + 1 + 16} = \frac{- 56}{26} = \frac{- 28}{13}

This gives:

a = 2 - \frac{84}{13} = \frac{- 58}{13}

b = 4 + \frac{28}{13} = \frac{80}{13}

c = 7 - \frac{112}{13} = \frac{- 21}{13}

Therefore:

2 a + b + 2 c = \frac{- 116 + 80 - 42}{13} = \frac{- 78}{13} = - 6

JEE Main · 2022 · 29 Jun 2022 · Shift 1 · Single correct option

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