### Planes $x-2y-2z+1=0$ & $2x-3y-6z+1=0$ ....

Let the acute angle bisector of the two planes $x-2y-2z+1=0$ and $2x-3y-6z+1=0$ be the plane P. Then which of the following points lies on P?

(A) $(0,2,-4)$
(B) $(4,0,-2)$
(C) $(-2,0,-\frac {1}{2})$
(D) $(3,1,-\frac {1}{2})$

Solution

Bisectors are given by,

$\frac{{x - 2y - 2z + 1}}{3} = \pm \frac{{2x - 3y - 6z + 1}}{7}$

$\Rightarrow 7x - 14y - 14z + 7 = \pm (6x - 9y - 18z + 3)$

Hence, $x-5y+4z+4=0$ & $13x-23y-32z+10=0$

Let $\theta$ be the angle between $x-2y-2z+1=0$ & $x-5y+4z+4=0$

$\cos \theta = \frac{{|1 + 10 - 8|}}{{3 \times \sqrt {42} }} = \frac{1}{{\sqrt {42} }} < \frac{1}{{\sqrt 2 }}$

$\theta > 45^\circ$

So, $x-5y+4z+4=0$ is the obtuse angle bisector.

$\therefore$ Acute angle bisector $P \equiv 13x - 23y - 32z + 10 = 0$

The point $(-2,0,-\frac {1}{2})$ satisfies P.