Finding the Equation of a Hyperbola Centered at (0,0) Given its Foci and Vertices

The vertices and foci are on the x-axis. Thus, the equation for the hyperbola will have the form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$

The vertices are $\left(\pm 6,0\right),$ so $a=6$ and $a^2=36.$

The foci are $\left(\pm 2\sqrt{10},0\right),$ so $c=2\sqrt{10}$ and $c^2=40.$

Solving for $b^2,$ we have

Finally, we substitute $a^2=36$ and $b^2=4$ into the standard form of the equation, $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$ The equation of the hyperbola is $\frac{x^2}{36}-\frac{y^2}{4}=1,$ as shown in Figure 6.