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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