CAT 2022 Slot 1 · Quant · Question 19

A rectangle inscribed in a circle has its diagonal as a diameter, since each angle of the rectangle is a right angle.

So with sides l and w, the diagonal is 2R and Pythagoras gives {l}^{2}+{w}^{2}=4{R}^{2}.

The perimeter is 2\left( l+w \right)=P, so l+w=\frac{P}{2}.

The area is lw, so the task is to get lw from these two facts.

Squaring the sum, {\left( l+w \right)}^{2}={l}^{2}+2lw+{w}^{2}.

Substituting both knowns, \frac{{P}^{2}}{4}=4{R}^{2}+2lw.

Rearranging, 2lw=\frac{{P}^{2}}{4}-4{R}^{2}.

Dividing by 2, lw=\frac{{P}^{2}}{8}-2{R}^{2}.

That is the second option.

Shortcut. A dimension check alone settles it: area must scale as a length squared, which rules out option 1, where {P}^{2} and PR are mixed inconsistently in magnitude. Testing a square of side \sqrt{2}R, where P=4\sqrt{2}R and the area is 2{R}^{2}, confirms option 2 and kills the rest.