Let the tangents at the points P and Q on the ellipse x22+y24=1 meet at the point R2, 22-2.If S is the focus of the ellipse on its negative major axis, then SP2+SQ2 is equal to

Equaation of Ellipse is x22+y24=1; e=12; S=0, -2 Chord of contact of R2,22-2 is x2+22-2y4=1 ⇒x2=1-2-1y2 solving with ellipse ⇒y=0, 2 ∴x=2, 1 P=1, 2 Q=2, 0 ∴SP2+SQ2=13

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Let the tangents at the points P and Q on the ellipse $\frac{x^{2}}{2} + \frac{y^{2}}{4} = 1$ meet at the point
$R (\sqrt{2}, 2 \sqrt{2} - 2) .$
If S is the focus of the ellipse on its negative major axis, then $S P^{2} + S Q^{2}$ is equal to

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answer is 13.

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Detailed Solution

$Equaation of Ellipse is \frac{x^{2}}{2} + \frac{y^{2}}{4} = 1; e = \frac{1}{\sqrt{2}}; S = (0, - \sqrt{2}) C h o r d o f c o n t a c t o f R (\sqrt{2}, 2 \sqrt{2} - 2) i s \frac{x}{\sqrt{2}} + \frac{(2 \sqrt{2} - 2) y}{4} = 1 \Rightarrow \frac{x}{\sqrt{2}} = 1 - \frac{(\sqrt{2} - 1) y}{2} solving with ellipse \Rightarrow y = 0, \sqrt{2} ∴ x = \sqrt{2}, 1 P = (1, \sqrt{2}) Q = (\sqrt{2}, 0) ∴ {(S P)}^{2} + {(S Q)}^{2} = 13$