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

Match the equation $\displaystyle 9x^2 - 25y^2 = 225$ 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 $



If we divide the equation by $225$, then we have $\displaystyle \frac{x^2}{25} - \frac{y^2}{9} = 1$.

Notice that the equation has the form $\displaystyle \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. Since the $x^2$-term is positive, then the hyperbola has a horizontal transverse axis; its vertices and foci are located on the $x$-axis. Its vertices is determined by $(\pm a, 0) \to (\pm 5, 0)$.

Therefore, it matches the graph I.