Question

A circle with centre $(2,3,0)$ and radius $1$ is drawn in the plane $z=0.$ Let the equation of the sphere which passes through this circle and the point $(1,1,1)$ be $x^2+y^2+z^2-kx-my-nz+p=0.$.Find $k+m+n-p$?

Answer 1

Equation of circle is given by, $(x-2{)}^2+(y-3{)}^2-1=0$ and plane is $z=0$
Thus equation of required sphere is given by,
$(x-2{)}^2+(y-3{)}^2-1+\lambda z=0$
Given this sphere passes through $(1,1,1)$
$\Rightarrow (1-2{)}^2+(1-3{)}^2-1+\lambda 1=0\to \lambda =-5$
Hence, the equation of sphere is,
$(x-2{)}^2+(y-3{)}^2-1-\lambda z=0$
$\Rightarrow x^2+y^2-4x-6y-5z+12=0$
$\Rightarrow k+m+n-p=3$