The Equation of a Hyperbola
If the given coordinates of the vertices and foci have the form (0, a) (0, -a), (0,c), and (0,-c) respectively, then the transverse axis is the y-axis.
The equation of this conic can be written in standard form:
[tex]\frac{y^2}{a^2}-\frac{x^2}{b^2}=1[/tex]
We are given the values of a=2, c = 5. We can find the value of b with the formula:
[tex]b^2=c^2-a^2[/tex]
Substituting values:
[tex]\begin{gathered} b=5^2-2^2 \\ \text{Calculating:} \\ b^2=21 \end{gathered}[/tex]
Thus the equation of the hyperbola is:
[tex]\frac{y^2}{4}-\frac{x^2}{21}=1[/tex]
The graph is shown below:
