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 \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) is \(x^2+y^2=a^2\)

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