If variable line y = kx + 2h is tangent to an ellipse 2x^2 + 3y^2 = 6 , then locus of P(h, k) is a c

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10-2. Parabola, Ellipse, Hyperbola

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If the variable line $y = kx + 2h$ is tangent to an ellipse $2x^2 + 3y^2 = 6$ , then locus of $P(h, k)$ is a conic $C$ whose eccentricity equals

A

$\frac{{\sqrt 5 }}{2}$

B

$\frac{{\sqrt 7 }}{3}$

C

$\frac{{\sqrt 7 }}{2}$

D

$\sqrt {\frac{7}{3}} $

Solution

By using condition of tangency,

we get $4 \mathrm{h}^{2}=3 \mathrm{k}^{2}+2$

$\therefore $ Locus of $P(h, k)$ is $4 x^{2}-3 y^{2}=2$ (which is hyperbola.)

Hence $e^{2}=1+\frac{4}{3} \Rightarrow e=\sqrt{\frac{7}{3}}$

Standard 11

Mathematics

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