Two vertical poles of height a and b subtend
the same angle $45°$ at a point on the line joining their
feet, the square of the distance between their tops is

Let$OP$ and$QR$ be the poles of height $a$ and
$b$respectively subtending angle 45° at $S$, a point between $O$
and $Q$. (Fig. 27.21) 

                          then $OS=a,QS=b$

                        $P{S}^2=2a^2$ and $R{S}^2=2b^2$

Since               $\mathrm{\angle }PSR={180}^{\circ }-1\left({45}^{\circ }+{45}^{\circ }\right)={90}^{\circ }$

                       $P{R}^2=P{S}^2+R{S}^2=2\left(a^2+b^2\right)$