Question

The equation of the plane passes through the point (2, 5, –3) perpendicular to the plane x + 2y + 2z = 1 and x – 2y + 3z = 4 is ______.

Options

• 10x – y – 4z = 27

• 3x – 4y + 2z – 20 = 0

• x + y + z = 5

• None of these

Answer 1

The equation of the plane passes through the point (2, 5, –3) perpendicular to the plane x + 2y + 2z = 1 and x – 2y + 3z = 4 is 10x – y – 4z = 27.

Explanation:

Given, the equation of plane passes through (2, 5, –3) is

a(x – 2) + b(y – 5) + c(z + 3) = 0  ...(i)

Which is perpendicular to the planes,

x + 2y + 2z = 1 and x – 2y + 3z = 4

Then, a + 2b + 2c = 0   ...(ii)

and a – 2b + 3c = 0  ...(iii)

Eliminating a, b, c from equations (i), (ii) and (iii), we get

`|(x - 2, y - 5, z + 3),(1, 2, 3),(1, -2, 3)|` = 0

`\implies` (x – 2) (6 + 4) – (y – 5) (3 – 2) + (z + 3) (–2 – 2) = 0

`\implies` 10x – y – 4z = 27