Question

Let $d$ is the distance of a plane $P$ from the point $\left(2,1,0\right)$ . Plane $P$ passes through $\left(1,2,1\right)$ and contains a line $x+2y+z=0=2x-y-z-1$ then $27d^2$ is

Answer 1

Answer: 50

Let equation of the plane is $\left(x+2y+z\right)+\lambda \left(2x-y-z-1\right)=0$
since it passes through $\left(1,2,1\right)$
$\Rightarrow 6+(-2)\lambda =0\Rightarrow \lambda =3$
$\therefore$ equation of plane $P$ is: $7x-y-2z-3=0$
$d=\left|\frac{14-1-3}{\sqrt{49+1+4}}\right|=\left|\frac{10}{\sqrt{54}}\right|$
$d^2=\frac{50}{27}$ ​​​​