Question

If foci of $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ coincide with the foci of $\frac{x^2}{25}+\frac{y^2}{9}=1$ and eccentricity of the hyperbola is 2, then :

• A. $a^2+b^2=16$
• B. there is no director circle to the hyperbola
• C. centre of the director circle is $(0,0)$
• D. length of latus rectum of the hyperbola $=12$

Answer 1

Answer: (A), (B), (D)

$a^2+b^2=16$
length of latus rectum of the hyperbola $=12$
there is no director circle to the hyperbola

$\frac{x^2}{25}+\frac{y^2}{9}=1$
which is an ellipse.
For ellipse, $a=5$ , $b=3$
$\Rightarrow e=\sqrt{\frac{25-9}{25}}=\frac{4}{5}$
$\therefore ae=4$
Hence, the foci are $(-4,0)$ and $(4,0)$.
For the hyperbola,
$ae=4,e=2$
$\Rightarrow a=2$
$b^2=4(4-1)=12$
$\Rightarrow b=\sqrt{12}$
$\Rightarrow a^2+b^2=16$
Since $b^2>a^2$, so there is no director circle to the hyperbola.
Length of latus rectum $=\frac{2b^2}{a}=12$