Question

The tangent and normal to the ellipse $x^2+4y^2=4$ at a point $P\left(\theta \right)$ on it meet the major axis in $Q$ and $R$ respectively. If $QR=2$, the eccentric angle $\theta$ of $P$ is given by

• A. $\cos \theta =\pm \frac{2}{3}$
• B. $\sin \theta =\pm \frac{2}{3}$
• C. $\tan \theta =\pm \frac{2}{3}$
• D. $\cot \theta =\pm \frac{2}{3}$

Answer 1

Answer: (A)

$\cos \theta =\pm \frac{2}{3}$

The equation of tangent to ellipse is $bxcos\theta +aysin\theta =ab$

It meets major axis at $x=\frac{a}{cos\theta }$

The equation of normal to ellipse is $axsec\theta -bycosec\theta =a^2-b^2$

It meets the major axis at $x=\frac{a^2-b^2}{asec\theta }$

Here $a=2$ and $b=1$

So, point $Q$ is $(\frac{2}{cos\theta },0)$ and point $R$ is $(\frac{3}{2sec\theta },0)$

The distance between them $QR=\frac{3cos\theta }{2}-\frac{2}{cos\theta }=2$

$\Rightarrow 3{\cos }^2\theta -4cos\theta -4=0$

$\Rightarrow cos\theta =-\frac{2}{3}$