Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

Let $P\left(x_0,y_0,z_0\right)$, the first line meet the x-axis at $A(a,0,0)$, the second meet the y-axis at $B(0,b,0)$, and the third pass through $C(0,0,c)$. Writing $r^2=x_0^2+y_0^2+z_0^2$, the conditions $PA\perp PB,PA\perp PC,PB\perp PC$ give $r^2-a{x}_0-b{y}_0=0,r^2-a{x}_0-c{z}_0=0,r^2-b{y}_0-c{z}_0=0$. Hence $a{x}_0=b{y}_0=c{z}_0=\frac{r^2}{2}$, and in particular $x_0^2+y_0^2+z_0^2-2c{z}_0=0$.