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

Explanation

Variable plane is passing through the point `A(a, b, c)` and foot of perpendicular from origin on this variable plane is `M(alpha, beta, gamma)`.
We have to find the locus of point `M`.
As shown in the figure, vectors `vec(AM) and vec(OM)` are perpendicular. Thus,
`" "vec(AM)*vec(OM)=0`
`rArr" "(alpha-a)alpha+(beta-b)beta+ (gamma-c)gamma=0`
Therefore, locus of `M` is
`" "x(x-a)+y(y-b)+z(z-c)=0`