Question

Find the equation for a hyperbola centered at (0, 0), with foci at (0,-sqrt73)) and (0,-sqrt73)) and vertices at (0, -8) and (0, 8).

Answer 1

Answer:

$\dfrac{y^2}{64}-\dfrac{x^2}{9}=1$ .

Step-by-step explanation:

Since vertices lie on y-axis. So, it is a vertical parabola of the form

$\dfrac{(y-k)^2}{a^2}-\dfrac{(x-h)^2}{b^2}=1$

where, (h,k) is center, $(h,k\pm c)$ is focus and $(h,k\pm a)$ is vertex.

Center is (0,0). So, h=0 and k=0.

Foci are $(0,\pm \sqrt{73})$. So $c=\sqrt{73}$.

Vertices are $(0,\pm 8)$. So $a=8$.

We know that,

$a^2+b^2=c^2$

$8^2+b^2=(\sqrt{73})^2$

$b^2=73-64$

$b=3$

Put h=0,k=0, a=8 and b=3 in equation (1).

$\dfrac{(y-0)^2}{8^2}-\dfrac{(x-0)^2}{3^2}=1$

$\dfrac{y^2}{64}-\dfrac{x^2}{9}=1$

Therefore, the required equation is $\dfrac{y^2}{64}-\dfrac{x^2}{9}=1$ .