Suppose $\mathrm{P}(-2,-3)$ and $\mathrm{Q}(3,7)$ are two points such that${\mathrm{PA}}^2+{\mathrm{QA}}^2$ is the least. Then the point $\mathrm{A}$ on the x-axis is

$\mathrm{P}\left(2,-3\right) \mathrm{Q}\left(3,7\right)$

Point A is lies on X-axis

$\mathrm{A}\left(\mathrm{x},0\right)$

Let $$\begin{gathered} \mathrm{S}={\mathrm{PA}}^2+{\mathrm{QA}}^2 \\ ={\left(\mathrm{x}+2\right)}^2+9+{\left(\mathrm{x}-3\right)}^2+49 \\ =2{\mathrm{x}}^2-2\mathrm{x}+71 \end{gathered}$$

S is min or max

${\mathrm{S}}^{\mathrm{I}}=0$

$\Rightarrow \mathrm{x}=2/4=1/2$

Point $\mathrm{A}=\left(1/2,0\right)$