Question

The plane x – y – z = 4 is rotated through 90° about its line of intersection with the plane x + y + 2z = 4. Find its equation in the new position.

Answer 1

Given planes are x – y – z = 4 ..... (1) 

and x + y + 2z = 4 ..... (2) 

Since the required plane passes through the line of intersection of planes (1) and (2) 

∴ its equation may be taken as 

x + y + 2z – 4 + k (x – y – z – 4) = 0 

or (1 + k)x + (1 – k)y + (2 – k)z – 4 – 4k = 0 ..... (3) 

Since planes (1) and (3) are mutually perpendicular, 

∴ (1 + k) – (1 – k) – (2 – k) = 0 

or, 1 + k – 1 + k – 2 + k = 0 

or, k = 2/3

Putting k = 2/3  in equation (3), we get, 

5x + y + 4z = 20 

This is the equation of the required plane.