Question

Suppose an ellipse and a hyperbola have the same pair of foci on the $x$-axis with centres at the origin and they intersect at $M(2,2)$. If the eccentricity of ellipse is $\frac{1}{2}$ and eccentricity of hyperbola is $\sqrt{\frac{m}{n}}$ where $m,n$ are coprime, then find the value of $(m+n)$.

Answer 1

Answer: 0010

Let $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $H:\frac{x^2}{A^2}-\frac{y^2}{B^2}=1$
$\Rightarrow \frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{4};e_E^2=1-\frac{b^2}{a^2}\Rightarrow \frac{b^2}{a^2}=\frac{3}{4}$
So, $a^2=\frac{28}{3}$ and $b^2=7$
$\Rightarrow E:\frac{x^2}{\frac{28}{3}}+\frac{y^2}{7}=1$
Now, slope of tangent at $M(2,2)$ on ellipse $=\frac{-3}{4}$
So, slope of tangent at $M(2,2)$ on hyperbola $\Rightarrow \frac{B^2}{A^2}=\frac{4}{3}$
As, $e_b^2=1+\frac{B^2}{A^2}\Rightarrow e_h{}^2=1+\frac{4}{3}=\frac{7}{3}$
So, $e_h=\sqrt{\frac{7}{3}}$. [Note that ellipse and hyperbola are confocal.].