Let P be an arbitrary point having the 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 x2+y2+z2=9, then the value of l−n is equal to:
Answer: 0
Solutions · 1
Official solution
\[\left(\frac{x+y+z}{\sqrt{3}}\right)^2+\left(\frac{lx-nz}{\sqrt{l^2+n^2}}\right)^2+\left(\frac{x-2y+z}{\sqrt{6}}\right)^2=9.\]
Expanding and matching with x2+y2+z2=9 (no cross term xz should remain other than what's consistent), the coefficient comparison forces
\[2ln=l^2+n^2 \implies (l-n)^2=0 \implies l-n=0.\]