<p>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:</p>
Step-by-Step Solution
Key Concept: The locus of intersection of tangents at corresponding points on the ellipse and auxiliary circle is the directrix of the ellipse.
<p>For a point P on the ellipse with eccentric angle \(\theta\), the tangent meets the auxiliary circle \(x^2 + y^2 = a^2\) at point Q. The tangent to the ellipse at P: \(\frac{x\cos\theta}{a} + \frac{y\sin\theta}{b} = 1\). The corresponding point on auxiliary circle is \(Q(a\cos\theta, a\sin\theta)\). The tangent at Q to the circle: \(x\cos\theta + y\sin\theta = a\). The intersection point of these two tangent lines lies on the directrix \(x = \frac{a^2}{c} = \frac{a}{e}\) where \(e\) is eccentricity, which is the line \(x = ae\) when properly derived.</p>
Correct Answer: A