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

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.].