JEE Main 2021 (Online) 26th February Evening ShiftQuestion 1
The triangle of maximum area that can be inscribed in a given circle of radius '$r$' is :
Differentiation and its Application
Solution
Concept. For a triangle in a circle of radius $r$, area $=2r^2\sin A\sin B\sin C$, which is largest when $A=B=C$. Formulas used. (1) $a=2r\sin A$ etc., so area $=\frac12ab\sin C=2r^2\sin A\sin B\sin C$(2) $\ln\sin$ is concave on $(0,\pi)$, so $\sin A\sin B\sin C$ is largest at $A=B=C=\frac\pi3$Step 1 - Maximum. By (1), (2), the triangle is equilateral. Step 2 - Side. $2r\sin60^\circ=\sqrt3\,r$. (Its height is $\frac{3r}{2}$, so (C) is wrong.) Answer: (D)