EllipseHardBloom L3
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 :
Options
A.$x = a/e$
B.$x = 0$
C.$y = 0$
D.none of these
Solution
Sol.(c) Equation of tangent at $P$ is
$$\begin{array}{r} \frac{xcos\theta}{a} + \frac{ysin\theta}{b} = 1\#(1) \end{array}$$
Equation of tangent at $Q$ is
$$\begin{array}{r} \frac{xcos\theta}{a} + \frac{ysin\theta}{a} = 1\#(2) \end{array}$$
(1) -(2)
$$\begin{matrix} & \ \Rightarrow \ ysin\theta\left( \frac{1}{b} - \frac{1}{a} \right) = 0 \\ & \ \Rightarrow \ y = 0 \end{matrix}$$
∴ Point of intersection lie on line $y = 0$
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