Question

A plane passes through a fixed point (a, b, c). Show that the locus of the foot of
the perpendicular to it from
the origin is the sphere,
x^2 + y^2 + z^2 - ax - by - cz = 0 .​

Answer 1

Let A(a,b,c) be the fixed point on the variable plane π and M(x,y,z) be the foot of the perpendicular from origin.

Now D.R's of OM are x−0,y−0,z−0 i.e x,y,z

D.R's of MA are x−a,y−b,z−c

Since OM⊥MA

x(x−a)+y(y−b)+z(z−c)=0⇒x2+y2+z2−ax−by−cz=0

Therefore radius =21a2+b2+c2