QJEEAdv23Mab3a2: JEE Advanced 2023 Mathematics - Class 11 JEEAdv23Mab3a2 Solution: Consider an obtuse-angled triangle A B C in which the…

Question

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Consider an obtuse-angled triangle

A B C

in which the difference between the largest and the smallest angle is

\frac{π}{2}

and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1.

Let a be the area of the triangle A B C . Then the value of (64 a)^{2} is

:

Accepted Answer

Area of

\triangle A B C=\frac{A B \cdot B C \cdot A C}{4 R}

\begin{aligned} \Rightarrow a & = \frac{8 \sin A \cdot \sin B \sin C}{4} \\ = 2 \sin (\frac{π}{2} - 2 C) \sin (\frac{π}{2} + C) \sin C \\ = 2 \cos 2 C \cdot \cos C \cdot \sin C \\ = \cos 2 C \cdot \sin 2 C \\ = \sqrt{1 - \sin^{2} 2 C} \cdot \sin 2 C \\ = \sqrt{1 - \frac{9}{6}} \cdot \times \frac{3}{4} \\ \Rightarrow a = \frac{3 \sqrt{7}}{16} \\ ∴ (64 a)^{2} = 1008 \end{aligned}

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