College Algebra, Chapter 8, 8.3, Section 8.3, Problem 6

Match the equation $\displaystyle y^2 - \frac{x^2}{9} = 1$ with the graphs labeled I-IV. Give reasons for your answers.

I.






$9x^2 - 25y^2 = 225$

II.







$16y^2 - x^2 = 144$

III.






$\displaystyle \frac{x^2}{4} - y^2 = 1$

IV.







$\displaystyle y^2 - \frac{x^2}{9} = 1 $



The equation has the form $\displaystyle \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$. Since the $y^2$-term is positive, the hyperbola has a vertical transverse axis; its vertices and foci are on the $y$-axis. Since $a^2 = 1$ and $b^2 = 9$, then we get $a = 1$ and $b = 3$. And if $c = \sqrt{a^2 + b^2}$, then $c = \sqrt{10}$.

Thus, the following are obtained

vertices $(0, \pm a) \rightarrow (0, \pm 1)$

foci $(0, \pm c) \rightarrow (0, \pm \sqrt{10})$

Therefore it matches the graph IV.