Question

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 ______.

Options

• 54

• 50

• –6

• –42

Answer 1

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 –6.

Explanation:

Given: The mirror image of (2, 4, 7) in the plane 3x – y + 4z = 2 is (a, b, c)

As we know, the mirror image of the point (x, y, z) in the plane ax + by + cz + d = 0 is given by,

`(x - x_1)/a = (y - y_1)/b = (z - z_1)/c = (-2(ax_1 + by_1 + cz_1 + d))/(a^2 + b^2 + c^2)`

⇒ `(a - 2)/3 = (b - 4)/(-1) = (c - 7)/4 = (-2(6 + (-4) + 28 - 2))/((3)^2 + (-1)^2 + (4)^2)`

⇒ `(a - 2)/3 = (b - 4)/(-1) = (c - 7)/4 = (-2(28))/26`

⇒ `(a - 2)/3 = (-28)/13, (b - 4)/1 = 28/13, (c - 7)/4 = (-28)/13`

⇒ a = `2 - 84/13`, b = `28/13 + 4`, c = `7 - 112/13`

⇒ a = `(-58)/13`, b = `80/13`, c = `(-21)/13`

⇒ 2a + b + 2c = `2((-58)/13) + 80/13 + 2((-21)/13)`

⇒ 2a + b + 2c = –6