Question

The equation of the hyperbola whose foci are $(6,5)$, $(-4,5)$ and eccentricity $5/4$ is

• A. $\frac{{\left(x-1\right)}^2}{16}-\frac{{\left(y-5\right)}^2}{9}=1$
• B. $\frac{x^2}{16}-\frac{y^2}{9}=1$
• C. $\frac{{\left(x-1\right)}^2}{16}-\frac{{\left(y-5\right)}^2}{9}=-1$
• D. None of these

Answer 1

Answer: (A)

Centre of the hyperbola is the mid point of the line joining the two foci, therefore, the coordinates of the centre are $(1,5)$. Now distance between the foci $=10$
$\Rightarrow 2ae=10$
$\Rightarrow ae=5$
$\Rightarrow a=4\left[\because e=5/4\right]$
Now, $b^2=a^2\left(e^2-1\right)$
$\Rightarrow b=3$
Hence, the equation of the hyperbola is
$\frac{{\left(x-1\right)}^2}{16}-\frac{{\left(y-5\right)}^2}{9}=1$.