Question

This hyperbola is centered at theorigin. Find its equation.Foci: (-2,0) and (2,0)Vertices: (-1,0) and (1,0)

Answer 1

SOLUTION

From the question, the center of the hyperbola is

$\begin{gathered} (h,k),\text{ which is } \\ (0,0) \end{gathered}$

a is the distance between the center to vertex, which is -1 or 1, and

c is the distance between the center to foci, which is -2 or 2.

b is given as

$\begin{gathered} b^2=c^2-a^2 \\ b^2=2^2-1^2 \\ b=\sqrt[]{3} \end{gathered}$

But equation of a hyperbola is given as

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

Substituting the values of a, b, h and k, we have

$\begin{gathered} \frac{(x-0)^2}{1^2}-\frac{(y-0)^2}{\sqrt[]{3}^2}=1 \\ \frac{x^2}{1}-\frac{y^2}{3}=1 \end{gathered}$

Hence the answer is

$\frac{x^2}{1}-\frac{y^2}{3}=1$