Question

Write the value of θ ϵ for which area of the triangle formed by points O (0, 0), A (a cos θ, b sin θ) and B (a cos θ, − b sin θ) is maximum.

Answer 1

Let A be the area of the triangle formed by the points O (0,0), A (acosθ,bsinθ) and B (acosθ,− bsinθ)

$$\begin{gathered} A=\frac{1}{2}\left|\begin{array}{ccc}0& 0& 1\\ a\mathrm{cos\theta }& b\mathrm{sin\theta }& 1\\ a\mathrm{cos\theta }& -b\mathrm{sin\theta }& 1\end{array}\right| \\ \Rightarrow A=\frac{1}{2}\left|\left(-ab\mathrm{sin\theta cos\theta }-ab\mathrm{sin\theta cos\theta }\right)\right| \\ \Rightarrow A=ab\mathrm{sin\theta cos\theta }=\frac{1}{2}\sin 2\mathrm{\theta } \end{gathered}$$
Now,
$$\begin{gathered} \therefore A_{max}=\frac{1}{2},\mathrm{when}\sin 2\mathrm{\theta }=1 \\ \Rightarrow \therefore A_{max}=\frac{1}{2},\mathrm{when}2\mathrm{\theta }=\frac{\mathrm{\pi }}{2}\Rightarrow \mathrm{\theta }=\frac{\mathrm{\pi }}{4} \end{gathered}$$

Hence, the area of the triangle formed by the given points is maximum when $\mathrm{\theta }=\frac{\mathrm{\pi }}{4}$.