Question

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

• A. $x=a / e$
• B. $x=0$
• C. $y=0$
• D. none of these

Answer 1

Answer: (C)

Tangent to the ellipse at point $P(a \cos \theta, b \sin \theta)$ is
$\frac{x}{a} \cos \theta+\frac{y}{b} \sin \theta=1$ ... (i)
Tangent to the circle at point $Q(a \cos \theta, a \sin \theta)$ is
$x \cos \theta+y \sin \theta=a$ ..... (ii)
Equations (i) and (ii) intersect at $(a / \cos \theta, 0)$ which lies on $y=0$.