JEE Main · 2019 · 09 Apr 2019 · Shift 2 · Circle

The two given vertices $(-8,5)$ and $(6,5)$ share the same height, so that side is horizontal with length $14$. The rectangle is inscribed in the circle, so the circle's centre is where the diagonals meet and lies on the given diameter line $3y=x+7$. The centre's $x$-coordinate is the midpoint of the horizontal side, $x=-1$; putting this on the line, $3y=-1+7=6$, so $y=2$ and the centre is $(-1,2)$. Since the centre's height is the average of the top and bottom sides, the other pair of vertices sits at $y=-1$, making the vertical side $|5-(-1)|=6$. The area is $14\times 6=84$ square units.