PQ is the chord joining the points and on the hyperbola

MATHEMATICS · Subjective · CBSE · Mathematics · Conic Sections · Conic Sections

Problem

PQ is the chord joining the points ϕ1\phi_1 and ϕ2\phi_2 on the hyperbola x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 . If ϕ1−ϕ2=2α\phi_1 - \phi_2 = 2\alpha , where α\alpha is constant, prove that PQ touches the hyperbola x2a2cos⁡2α−y2b2=1.\frac{x^2}{a^2} \cos^2 \alpha - \frac{y^2}{b^2} = 1.

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