JEE 2022 — Mathematics PYQ

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Question

JEE 2022 — Mathematics PYQ

JEE | Mathematics | 2022

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:

Choose the correct answer:

Explanation

Solution

1. Use the Mirror Image Formula:

For a point $(x_1, y_1, z_1)$ and a plane $Ax + By + Cz + D = 0$ , the image $(a, b, c)$ is given by:

\frac{a - x_1}{A} = \frac{b - y_1}{B} = \frac{c - z_1}{C} = -2 \frac{Ax_1 + By_1 + Cz_1 + D}{A^2 + B^2 + C^2}

2. Plug in the Values:

Point is $(2, 4, 7)$ and plane is $3x - y + 4z - 2 = 0$ .

\frac{a - 2}{3} = \frac{b - 4}{-1} = \frac{c - 7}{4} = -2 \frac{3(2) - (4) + 4(7) - 2}{3^2 + (-1)^2 + 4^2}

\frac{a - 2}{3} = \frac{b - 4}{-1} = \frac{c - 7}{4} = -2 \frac{6 - 4 + 28 - 2}{9 + 1 + 16}

\frac{a - 2}{3} = \frac{b - 4}{-1} = \frac{c - 7}{4} = -2 \left( \frac{28}{26} \right) = -\frac{28}{13}

3. Solve for $a, b,$ and $c$ :

$a = 2 + 3(-\frac{28}{13}) = 2 - \frac{84}{13} = -\frac{58}{13}$
$b = 4 - 1(-\frac{28}{13}) = 4 + \frac{28}{13} = \frac{80}{13}$
$c = 7 + 4(-\frac{28}{13}) = 7 - \frac{112}{13} = -\frac{21}{13}$

4. Calculate $2a + b + 2c$ :

2\left(-\frac{58}{13}\right) + \left(\frac{80}{13}\right) + 2\left(-\frac{21}{13}\right)

= \frac{-116 + 80 - 42}{13} = \frac{-78}{13} = -6

Correct Option: (C) -6

Choose the correct answer:

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