Question

The points $P,Q,R$ with eccentric angles $\theta ,\theta +\alpha ,\theta +2\alpha$ where $\alpha \in (0,\pi )$ are taken on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ then

• A. area of $\Delta PQR$ is independent of $\theta$ only
• B. area of $\Delta PQR$ is independent of $\alpha$ only
• C. area of $\Delta PQR$ is independent of both $\theta$ and $\alpha$
• D. area of $\Delta PQR$ is maximum when $\alpha =\frac{2\pi }{3}$

Answer 1

Answer: (A), (D)

The correct options are
A area of $\Delta PQR$ is independent of $\theta$ only
D area of $\Delta PQR$ is maximum when $\alpha =\frac{2\pi }{3}$

Equation of the ellipse is,
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$

Given,
$P\left(\theta \right),Q(\theta +\alpha ),R(\theta +2\alpha )$ are the points, which lie on the ellipse.

$A=\text{Area of}△PQR\Rightarrow A=\frac{1}{2}\left|\begin{array}{ccc}a\cos \theta & b\sin \theta & 1\\ a\cos (\theta +\alpha )& b\sin (\theta +\alpha )& 1\\ a\cos (\theta +2\alpha )& b\sin (\theta +2\alpha )& 1\end{array}\right|\Rightarrow A=\frac{ab}{2}\left[\right\{\cos (\theta +\alpha )\sin (\theta +2\alpha )-\sin (\theta +\alpha )\cos (\theta +2\alpha )\}-\left\{\cos \right(\theta \left)\sin \right(\theta +2\alpha )-\sin (\theta \left)\cos \right(\theta +2\alpha \left)\right\}+\left\{\cos \right(\theta \left)\sin \right(\theta +\alpha )-\sin (\theta \left)\cos \right(\theta +\alpha \left)\right\}]\Rightarrow A=\frac{ab}{2}(\sin \alpha -\sin 2\alpha +\sin \alpha )\Rightarrow A=ab\sin \alpha (1-\cos \alpha )$

Which is independent of $\theta$

Now, $\frac{dA}{d\alpha }=ab(\cos \alpha -\cos 2\alpha )$
For maxima, $\frac{dA}{d\alpha }=0$
$\Rightarrow \cos \alpha -\cos 2\alpha =0\Rightarrow \cos 2\alpha =\cos \alpha \Rightarrow 2\alpha =2n\pi \pm \alpha \Rightarrow \alpha =2n\pi ,3\alpha =2n\pi \therefore \alpha =\frac{2\pi }{3}[\because \alpha \in (0,\pi \left)\right]$