Question

Write the value of θ ϵ \[\left( 0, \frac{\pi}{2} \right)\] 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)

\[A = \frac{1}{2}\begin{vmatrix}0 & 0 & 1 \\ acos\theta & bsin\theta & 1 \\ acos\theta & - bsin\theta & 1\end{vmatrix}\]

\[ \Rightarrow A = \frac{1}{2}\left| \left( - absin\theta cos\theta - absin\theta cos\theta \right) \right|\]

\[ \Rightarrow A = absin\theta cos\theta = \frac{1}{2}\sin2\theta\]

Now,

\[\therefore A_{\text { max }} = \frac{1}{2}, \text { when  }\sin2\theta = 1\]

\[ \Rightarrow \therefore A_{\text { max }} = \frac{1}{2},\text {  when } 2\theta = \frac{\pi}{2} \Rightarrow \theta = \frac{\pi}{4}\]

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