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 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, $a=\frac{1}{\sqrt{2}},b=\sqrt{\frac{2}{3}},\left(a>b\right)$
$\Rightarrow e=\sqrt{1+\frac{b^2}{a^2}}=\sqrt{1+\frac{4}{3}}\Rightarrow e=\sqrt{\frac{7}{3}}$