If tangents to the parabola y2-4 intersect the ellipse x2-a2-y2-b2-1 at A and B- then find the locus of point of intersection of tangents at A and B-

#10;Let #10;P = (h, k) be the point of intersection of tangents #160;at A amp; B #10; ∴ #160;equation of chord of contact AB #10;is #160; ...

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Byju's Answer

Standard XII

Mathematics

Parametric Form of Tangent: Hyperbola

If tangents t...

Question

If tangents to the parabola $y^{2} = 4$ intersect the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ at A and B, then find the locus of point of intersection of tangents at A and B.

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Solution

Let P = (h, k) be the point of intersection of tangents at A & B $∴$ equation of chord of contact AB is $\frac{x h}{a^{2}} + \frac{y k}{b^{2}} = 1$ Which touches the paraboia equation of tangent to parabola $y^{2} = 4 a x y = m x + \frac{a}{m} \Rightarrow m x - y = - \frac{a}{m}$ . Equation (i) & (ii) as must be same

$∴ \frac{m}{(\frac{h}{a^{2}})} = \frac{- 1}{(\frac{k}{b^{2}})} = \frac{- \frac{a}{m}}{1} \Rightarrow m = - \frac{h}{k} \frac{b^{2}}{a^{2}}$ & $m = \frac{a k}{b^{2}} ∴ - \frac{h b^{2}}{k a^{2}} = \frac{a k}{b^{2}} \Rightarrow$ locus of P is $y^{2} = - \frac{b^{4}}{a^{3}} . x$

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If tangents to the parabola $y^{2} = 4 a x$ intersect the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ at A and B, then find the locus of point of intersection of tangents at A and B.

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A

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Q. Tangents one to each of the ellipses

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and

\frac{x^{2}}{a^{2} + λ} + \frac{y^{2}}{b^{2} + λ} = 1

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Q. If

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constant, then the locus of the point of intersection of tangents at

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\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

p (θ), D (θ + \frac{π}{2})

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If tangents to the parabola y2=4 intersect the ellipse x2a2+y2b2=1 at A and B, then find the locus of point of intersection of tangents at A and B.

If tangents to the parabola $y^{2} = 4$ intersect the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ at A and B, then find the locus of point of intersection of tangents at A and B.