# Eccentric Ellipse

Geometry Level 3

The tangent at $$P(\phi)$$ of the ellipse $$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$$ meets its auxiliary circle at points $$Q$$ and $$R$$. If the chord $$QR$$ subtends a right angle at the origin, find the value of:

$e\sqrt{1+\sin^2 \phi}$

Details and Assumptions

• Assume $$a > b$$
• $$P(\phi)$$ refers to the point $$P(a \cos \phi, b\sin \phi)$$, where $$\phi$$ is the eccentric angle
• The auxiliary circle of the ellipse is the circumcircle of the ellipse.
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