F and \( Q \) are two points on the ellipse \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) whos...

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F and \( Q \) are two points on the ellipse \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) whose eccentric angles are differ by \( 90^{\circ} \), then:
(a) Locus of point of intersection of tangents at \( P \) and \( Q \) is \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=2 \)
(b) Locus of mid-point \( (P, Q) \) is \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=\frac{1}{2} \)
(c) Product of slopes of \( O P \) and \( O Q \) where \( O \) is the centre is \( \frac{-b^{2}}{a^{2}} \)
(d) Max. area of \( \triangle O P Q \) is \( \frac{1}{2} \mathrm{ab} \)
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