Three straight lines mutually perpendicular to each other meet in a point P and one of them intersects the x-axis and another intersects the y-axis, while the third line passes through a fixed point (0, 0, c) on the z-axis. Then the locus of P is

Let  $L_1,L_2,L_3$ be the mutually perpendicular lines and $P\left(x_0,y_0,z_0\right)$ be their point of concurrence. If $L_1$ cuts the x-axis at $A(a,0,0),L_2$ meets the y-axis at $B(0,b,0)$ and $C(0,0,c)\in L_3,$ then $L_{1}11\left(x_0-a,y_0,z_0\right),L_{2}11\left(x_0,y_0-b,z_0\right).$

$Hence \left.\begin{array}{l}{x}_0\left(x_0-a\right)+y_0\left(y_0-b\right)+z_0^2=0\\ x_0^2+\left(y_0-b\right)y_0+z_0\left(z_0-c\right)=0\end{array}\right\}$

$x_0\left(x_0-a\right)+y_0^2+z_0\left(z_0-c\right)=0$

Eliminating a and b from the equations, we get

$x_0^2+y_0^2+z_0^2-2c{z}_0=0$