Question

If the planes x – cy – bz = 0, cx – y + az = 0 and bx + ay – z = 0 pass through a straight line, then find the value of a² + b² + c² + 2abc.

Answer 1

Given planes are x – cy – bz = 0 ..... (1) 

cx – y + az = 0 ..... (2) 

bx + ay – z = 0 ..... (3) 

Equation of any plane passing through the line of intersection of planes (1) and (2) may be taken as 

x – cy – bz + λ (cx – y + az) = 0 

or, x (1 + λc) – y (c + λ) + z (– b + aλ) = 0..... (4) 

If planes (3) and (4) are the same, then equations (3) and (4) will be identical.