Question

The equation of the plane through the point $(2,5,-3)$ perpendicular to the planes $x+2y+2z=1$ and $x-2y+3z=4$ is

• A. $3x-4y+2z-20=0$
• B. $7x-y+5z=30$
• C. $x-2y+z=11$
• D. $10x-y-4z=27$

Answer 1

Answer: (D)

Equation of plane passing through $(2,5,-3)$ is
$a(x-2)+b(y-5)+c(z+3)=0\dots (1)$
which is perpendicular to
$x+2y+2z=1$ and $x-2y+3z=4$
then $a+2b+2c=0\dots (2)$
and $a-2b+3c=0\dots (3)$
On solving $(1)$, $(2)$ and $(3)$, we get
$\left|\begin{array}{ccc}x-2& y-5& z+3\\ 1& 2& 2\\ 1& -2& 3\end{array}\right|=0$
$\Rightarrow 10x-y-4z-27=0$