Question

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

Answer 1

Line $y=kx+2h---\left(i\right)$

Slope $m=k$ and constant $c_1=2h$

Ellipse:$2x^2+3y^2=6$

$\Rightarrow \frac{x^2}{3}+\frac{y^2}{2}=1---\left(ii\right)$

If $\left(i\right)$ is tangent to $\left(ii\right)$

$\Rightarrow c_1=\sqrt{a^2{m}^2+b^2}$

$\Rightarrow 4h^2=3k^2+2$

$\Rightarrow 4h^2-3k^2=2$

$\Rightarrow \frac{h^2}{\frac{1}{2}}-\frac{k^2}{\frac{2}{3}}=1$

For locus of $P(h,k)$, replacing $h\to x,k\to y$

$\Rightarrow \frac{x^2}{\frac{1}{2}}-\frac{y^2}{\frac{2}{3}}=1$ is a hyperbola(conic)

so, eccentricity of this conic $e=\sqrt{1+\frac{\left(\frac{2}{3}\right)}{\left(\frac{1}{2}\right)}}\Rightarrow e=\sqrt{\frac{7}{3}}$

$3e^2=\frac{7}{3}.3=7$