Question

Graphs of Hyperbolas

Match the equation with the graphs labeled I–IV. Give reasons for your answers.

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

Short answer

The graph of hyperbola matched with diagram labeled as $I$ is

Step-by-step solution

Step 1. Given information

Equation of an hyperbola is given as $9x^2-25y^2=225$.

Step 2. Concept used

The standard form of a hyperbola with a center at the origin and with a vertical transverse axis is

$\frac{y^2}{a^2}-\frac{x^2}{b^2}$ where $a>0,b>0$.

The vertices are $(0,\pm a)$.

The foci are $(0,\pm c)$ with $c^2=a^2+b^2$

We will convert the given equation of the hyperbola in the standard form and then by comparing we will get the value of $a,b$.

Step 3. Calculation

Writing the given hyperbola equation in standard form we get,

$\begin{array}{l}9x^2-25y^2=225\\ \Rightarrow \frac{9}{225}{x}^2-\frac{25}{225}{y}^2=\frac{225}{225}\\ \Rightarrow \frac{x^2}{5^2}-\frac{y^2}{3^2}=1\end{array}$

On comparing the giving hyperbola equation with the standard hyperbola equation we get,

$\begin{array}{l}a=5\\ b=3\end{array}$

Hence the vertices are $(\pm 5,0)$

And the transverse axis is horizontal

Therefore graph labeled as $1$ is the correct answer.