If the chord joining the points - and - on the ellipse x2-a2 - y2-b2 - 1 subtends a right angle at a- 0- then tan -2 tan -2 -

P(a cosα, b sinα), Q = (a cosβ, b sinβ), R = (a, 0) Since PR⊥QR then b sinα/a(cosα 1). b sinβ/a(cosβ 1) = 1 ⇒b^2/a^2.α/2.β/2 = 1 ⇒tanα/2tanβ/2 = b^2/a^2 ...

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Standard XIII

Mathematics

Parametric Representation: Ellipse

If the chord ...

Question

If the chord joining the points $α$ and $β$ on the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ subtends a right angle at (a, 0), then $tan (\frac{α}{2}) tan (\frac{β}{2}) =$

\frac{a^{2}}{b^{2}}

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\frac{b^{2}}{a^{2}}

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- \frac{a^{2}}{b^{2}}

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- \frac{b^{2}}{a^{2}}

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Solution

The correct option is D $- \frac{b^{2}}{a^{2}}$
$P (a cos α, b sin α), Q = (a cos β, b sin β), R = (a, 0)$
Since PR⊥QR then $\frac{b sin α}{a (cos α - 1)} . \frac{b sin β}{a (cos β - 1)} = - 1 \Rightarrow \frac{b^{2}}{a^{2}} . cot \frac{α}{2} . cot \frac{β}{2} = - 1 \Rightarrow tan \frac{α}{2} tan \frac{β}{2} = - \frac{b^{2}}{a^{2}}$

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