Question

If planes x – cy – bz = 0, cx – y + az = 0 and bx + ay – z = 0 pass through a straight line then a² + b² + c² = ______.

Options

• 1 – abc

• abc – 1

• 1 – 2abc

• 2abc – 1

Answer 1

If planes x – cy – bz = 0, cx – y + az = 0 and bx + ay – z = 0 pass through a straight line then a² + b² + c² = 1 – 2abc.

Explanation:

Planes x – cy – bz = 0

cx – y + az = 0

bx + ay – z = 0

Since the given planes pass through a straight line.

∵ Planes are concurrent

`|(1, -c, -b),(c, -1, a),(b, a, -1)|` = 0

1(1 – a²) + c(–c – ab) – b(ac + b) = 0

1 – a² – c² – abc – abc – b² = 0

a² + b² + c² + 2abc = 1

a² + b² + c² = 1 – 2abc