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Equation of Tangent in Point Form

If the variab...

Question

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

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Solution

By using the tangency of the ellipse
$4 h^{2} = 3 k^{2} + 2$
$∴$ locus of the $P (h, k)$ is
$4 x^{2} - 3 y^{2} = 2$
It is an equation of the hyperbola
$e^{2} = \frac{7}{3}$

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