Question

Let a plane $P$ pass through the point $(3,7,-7)$ and contain the line, $\frac{x-2}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$. If distance of the plane $P$ from the origin is $d,$ then $d^2$ is equal to

Answer 1

Equation of plane through the point $(3,7,-7)$ and containing the line $\frac{x-2}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$ is
$\left|\begin{array}{ccc}x-2& y-3& z+2\\ 3-2& 7-3& -7+2\\ -3& 2& 1\end{array}\right|=0$
$\Rightarrow \left|\begin{array}{ccc}x-2& y-3& z+2\\ 1& 4& -5\\ -3& 2& 1\end{array}\right|=0$
$\Rightarrow x+y+z-3=0$
Now, distance of the plane from origin is
$d=\left|\frac{-3}{\sqrt{1^2+1^2+1^2}}\right|$
$\therefore d^2=3$