Tangents from a point $P$ on the auxiliary circle of $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ are drawn to the ellipse. The chord of contact of the ellipse with respect to $P$ always passes through a fixed point. That fixed point is:
Step-by-Step Solution
Key Concept: Let $P = (a \cos \theta, a \sin \theta)$ on auxiliary circle $x^2 + y^2 = a^2$. Chord of contact to ellipse: $x \cos \theta/a + y \sin \theta/b^2 \cdot a = 1...$ The chord of contact from $P$ is $T = 0$, which corresponds to the polar of $P$ w.r.t. the ellipse — it passes through the pole of the line $OP$, landing on the corresponding directrix foot.
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Correct Answer: (1)