Question

The equation of a plane bisecting the angle between the plane $2x-y+2z+3=0$ and $3x-2y+6z+8=0$ is/are:

• A. $5x-y-4z-3=0$
• B. $5x-y-4z-45=0$
• C. $23x-13y+32z+5=0$
• D. $23x-13y+32z+45=0$

Answer 1

Answer: (A), (D)

$23x-13y+32z+45=0$

Given planes $2x-y+2z+3=0,3x-2y+6z+8=0$
Equation of the plane bisecting the two planes $a_{1}x+b_{1}y+c_{1}z+d_1=0$ and $a_{2}x+b_{2}y+c_{2}z+d_2=0$ is obtained by
$\Rightarrow \frac{a_{1}x+b_{1}y+c_{1}z+d_1}{\sqrt{a_1^2+b_1^2+c_1^2}}=\pm \frac{a_{2}x+b_{2}y+c_{2}z+d_2}{\sqrt{a_2^2+b_2^2+c_2^2}}$
$\Rightarrow \frac{2x-y+2z+3}{\sqrt{4+1+4}}=\pm \frac{3x-2y+6z+8}{\sqrt{9+4+36}}$
$\Rightarrow \frac{2x-y+2z+3}{3}=\pm \frac{3x-2y+6z+8}{7}$
Case $I$:
$7(2x-y+2z+3)=3(3x-2y+6z+8)$
We get: : $5x-y-4z-3=0$
Case $II$:
$7(2x-y+2z+3)=-3(3x-2y+6z+8)$
We get: $23x-13y+32z+45=0$