Question

If $A$ and $B$ be the points $(3,4,5)$ and $(-1,3,-7)$ respectively. Find the equation of the set of points $P$ such that $P{A}^2$ +$P{B}^2$ $=K{}^2$, where $K$ is a constant

Answer 1

Given coordinates of points$A$ and $B$ are $(3,4,5)$ and $(-1,3,-7)$ respectively.

Let the coordinates of point $P$ be $(x,y,z)$.

Given, $P{A}^2+P{B}^2=K^2$
$\Rightarrow (x-3{)}^2+(y-4{)}^2+(z-5{)}^2+(x+1{)}^2+(y-3{)}^2+(z+7{)}^2=K^2$

$x^2-6x+9+y^2-8y+16+z^2-10z+25+x^2+2x+1+y^2-6y+9+z^2+14z+49=K^2$

$\Rightarrow 2x^2+2y^2+2z^2-4x-14y+4z+109=K^2$

$\Rightarrow 2(x^2+y^2+z^2-2x-7y+2z)=K^2-109$

$\Rightarrow x^2+y^2+z^2-2x-7y+2z=\frac{K^2-109}{2}$

which is the required equation.