Question

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

• A. $a^2+b^2=14$
• B. There is a director circle of the hyperbola
• C. Centre of the director circle is $(0,0)$
• D. Length of latus rectum of the hyperbola is $12$

Answer 1

Answer: (D)

Length of latus rectum of the hyperbola is $12$

For the ellipse, $a=5,b=3$ and $e=\sqrt{\frac{25-9}{25}}=\frac{4}{5}$
$\therefore ae=4$
Hence, the focii are $(-4,0)$ and $(4,0)$

As the focii of ellipse and hyperbola coincide
For the hyperbola, $ae=4,e=2$
$\therefore a=2$
$b^2=a^2(e^2-1)=4(4-1)=12$
$\therefore b=\sqrt{12}$

Length of latus rectum
$=\frac{2b^2}{a}=2\times \frac{12}{2}$
$=12$

Hence, option D is correct.