Mathematics - Point Question with Solution | TestHub

Let a straight line through the vertex $P$ of a given $△PQR$ intersects the side $QR$ at the point $S$ and the circumcircle of $△PQR$ at the point $T$. Points $P,Q,R,T$ are concyclic, hence $$\begin{array}{rlrl}PS\cdot ST& =QS\cdot SR& & \\ \Rightarrow P_1& \ge 2\sqrt{PS\cdot ST}& & \\ & \text{ Now, }\frac{PS+ST}{2}& \ge \sqrt{PS\cdot ST}& [\because \mathrm{AM}\ge \mathrm{GM}]\end{array}$$
and $\frac{1}{PS}+\frac{1}{ST}\ge \frac{2}{\sqrt{PS\cdot ST}}=\frac{2}{\sqrt{QS\cdot SR}}$ Also, $\frac{SQ+QR}{2}\ge \sqrt{SQ\cdot SR}$ $\Rightarrow \frac{QR}{2}\ge \sqrt{SQ\cdot SR}$ $$\begin{array}{ll}\Rightarrow & \frac{1}{\sqrt{SQ\cdot SR}}\ge \frac{2}{QR}\Rightarrow \frac{2}{\sqrt{SQ\cdot SR}}\ge \frac{4}{QR}\\ \text{ Hence, }& \frac{1}{PS}+\frac{1}{ST}\ge \frac{2}{\sqrt{QS\cdot SR}}\ge \frac{4}{QR}.\end{array}$$