Question

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

• A. $\frac{\sqrt{5}}{2}$
• B. $\frac{\sqrt{7}}{3}$
• C. $\frac{\sqrt{7}}{2}$
• D. $\sqrt{\frac{7}{3}}$

Answer 1

Answer: (D)

We have, the equation of an ellipse $2x^{2}+3y^{2}=6.$
And given variable line tangent to an ellipse is $y=kx+2h.$
By using the condition of tangency, we get $4h^{2}=3k^{2}+2.$
$\Rightarrow 4h^{2}-3k^{2}=2$
then,the locus of the point $P\left(h , k\right)$ is the hyperbola $4x^{2}-3k^{2}=2.$
$\Rightarrow \frac{x^{2}}{\left(\frac{1}{\sqrt{2}}\right)^{2}}-\frac{y^{2}}{\left(\sqrt{\frac{2}{3}}\right)^{2}}=1$
Here, $a=\frac{1}{\sqrt{2}},b=\sqrt{\frac{2}{3}},\left(a > b\right)$
$\Rightarrow e=\sqrt{1 + \frac{b^{2}}{a^{2}}}=\sqrt{1 + \frac{4}{3}}\Rightarrow e=\sqrt{\frac{7}{3}}$