Question

Find the equation of the sphere having centre on the line $2x-3y=0,5y+2z=0$ and passing through the points $(0,-2,-4)$ and $(2,1,-1)$

Answer 1

$\begin{array}{c}Lettheequationofspherebe\\ x^2+y^2+z^2+2ux+2vy+2wz+e=0\\ since\left(0,-2,4\right)and\left(2,-1,-1\right)lieonthesphere.\\ 0^2+4+16+0-4v+8w+e=0\\ -4v-2w+e=-\mathrm{20..............}\left(1\right)\\ frompoint\left(2,-1,-1\right)\\ 4u-2v-2w+e=-\mathrm{6...............}\left(2\right)\\ Centreofsphere\left(-u,-v,-w\right)liesonthelinesso,\\ 5v+2w=\mathrm{0............}\left(3\right)\\ 2u-3v=\mathrm{0.................}\left(4\right)\\ fromequation\left(1\right),\left(2\right),\left(3\right)\&\left(4\right)\\ u=-3,v=-2,w=5,e=12\\ so,equationis\\ x^2+y^2+z^2-6x-4y+10z+12=0\end{array}$