Embibe Experts Solutions for Chapter: Circle, Exercise 1: IOQM - PRMO and RMO 2018

Let $R$ denote the circular region in the $xy-$ plane bounded by the circle $x^2+y^2=36$. The lines $x=4$ and $y=3$ divide $R$ into four regions $R_i, i=1, 2, 3, 4$. If $\left|R_i\right|$ denotes the area of the region $R_i$ and if $\left|R_1\right|>\left|R_2\right|>\left|R_3\right|>\left|R_4\right|,$ determine $\left|R_1\right|-\left|R_2\right|-\left|R_3\right|+\left|R_4\right|$. (Here $\left|\Omega \right|$ denotes the area of the region $\mathrm{\Omega }$ in the plane)

In parallelogram $ABCD\mathit{,}\mathit{ }AC=10$ and $BD=28.$ The points $K$ and $L$ in the plane of $ABCD$ move in such a way that $AK=BD$ and $BL=AC.$ Let $M$ and $N$ be the midpoints of $CK$ and $DL\mathit{,}$ respectively. What is the maximum value of ${\cot }^2\left(\frac{\angle BMD}{2}\right)+{\tan }^2\left(\frac{\angle ANC}{2}\right)$?

Let $ABCD$ be a trapezium in which $AB\parallel CD$ and $AD\perp AB$. Suppose $ABCD$ has an incircle which touches $AB$ at $Q$ and $CD$ at $P$. Given that $PC=36$ and $QB=49,$ find $PQ.$