Let the mirror image of the point $(a, b, c)$ with respect to the plane $3x-4y+12z+19=0$ be $(a-6, \beta, \gamma)$. If $a+b+c=5$, then $7\beta-9\gamma$ is equal to ______.

JEE Main · 2022 · 27 Jun 2022 · Shift 1 · Three Dimensional Geometry

Let the mirror image of the point

(a, b, c)

with respect to the plane

3 x - 4 y + 12 z + 19 = 0

(a - 6, β, γ)

. If

a + b + c = 5

, then

7 β - 9 γ

is equal to ______.

Answer: 137

The midpoint

M

of the point

(a, b, c)

and its mirror image

(a - 6, β, γ)

is:

M = (a - 3, \frac{b + β}{2}, \frac{c + γ}{2})

Since

M

lies on the plane

3 x - 4 y + 12 z + 19 = 0

3 (a - 3) - 4 (\frac{b + β}{2}) + 12 (\frac{c + γ}{2}) + 19 = 0

\Rightarrow 6 a - 4 b + 12 c - 4 β + 12 γ + 20 = 0

... (1)

The line joining the point and its image is perpendicular to the plane, so:

\frac{a - ( a - 6 )}{3} = \frac{b - β}{- 4} = \frac{c - γ}{12}

\frac{6}{3} = \frac{b - β}{- 4} = \frac{c - γ}{12}

This gives:

β = b + 8

and

γ = c - 24

.

From

a + b + c = 5

a + (β - 8) + (γ + 24) = 5 \Rightarrow a = - β - γ - 11

Substituting into (1):

6 (- β - γ - 11) - 4 (β - 8) + 12 (γ + 24) - 4 β + 12 γ + 20 = 0

- 6 β - 6 γ - 66 - 4 β + 32 + 12 γ + 288 - 4 β + 12 γ + 20 = 0

- 14 β + 18 γ + 274 = 0

7 β - 9 γ = 137

JEE Main · 2022 · 27 Jun 2022 · Shift 1 · Integer answer