Question

The point of intersection of the tangents at the point $P$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(a>b)$ and its corresponding point $Q$ on the auxiliary circle, meet on the line

• A. $x=\frac{a}{e}$
• B. $x=0$
• C. $y=0$
• D. $x=-\frac{a}{e}$

Answer 1

The correct option is C $y=0$


Tangent to the ellipse at point $P(a\cos \theta ,b\sin \theta )$ is
$\frac{x}{a}\cos \theta +\frac{y}{b}\sin \theta =1\dots \left(1\right)$

Tangent to the circle at point $Q(a\cos \theta ,a\sin \theta )$ is $x\cos \theta +y\sin \theta =a...\left(2\right)$
The point of intersection of lines $\left(1\right)$ and $\left(2\right)$ is $(\frac{a}{\cos \theta },0),$ which lies on $y=0.$