PART - 2 CH - 11 Introduction to Three Dimensional Geometry — MATHS STD 11 Science — Question

Let $A (1,5,7)$ and $B (5,1,-4)$ and the required point on $Z-\text {axis}$ be $P (0,0, z)$.
$\therefore\quad$ According to question,
$AP = BP$
$\Rightarrow \quad AP ^2= BP ^2$
$\begin{aligned} \Rightarrow(1-0)^2+(5 & -0)^2+(7-z)^2 =(5-0)^2+(1-0)^2+(-4-z)^2\end{aligned}$
$\Rightarrow 1+25+49+z^2-14 z=25+1+16+z^2+8 z$
$\Rightarrow \quad 49-16=8 z+14 z$
$\Rightarrow \quad 33=22 z$
$\Rightarrow \quad z=\frac{33}{22}=\frac{3}{2}$
Hence, required point on $Z-\text {axis}$ is $(0,0, z)$
$=\left(0,0, \frac{3}{2}\right)$