The tangent at $(a\cos\theta, b\sin\theta)$ on the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ meets the auxiliary circle in two points. The chord joining them subtends a right angle at the centre. Find the eccentricity of the ellipse.
Answer I got is $\dfrac{1}{\sqrt{1+\sin^2\theta}}$. Is it correct?