Question

Let $P$ be an arbitrary point having sum of the squares of the distances from the planes $x+y+z=0,lx-nz=0$ and $x-2y+z=0,$ equal to $9$. If the locus of the point $P$ is $x^2+y^2+z^2=9,$ then the value of $l-n$ is equal to

Answer 1

Let the point $P=(x,y,z)$
Now, according to the given condition, we get
${\left(\frac{x+y+z}{\sqrt{3}}\right)}^2+{\left(\frac{lx-nz}{\sqrt{l^2+n^2}}\right)}^2+{\left(\frac{x-2y+z}{\sqrt{6}}\right)}^2=9$
As the given locus is $x^2+y^2+z^2=9$, so the coefficient of $xz$ is
$\frac{2}{3}-\frac{2ln}{l^2+n^2}+\frac{2}{6}=0\Rightarrow \frac{2ln}{l^2+n^2}=1\Rightarrow (l-n{)}^2=0\therefore l-n=0$