Question

Graphs of Hyperbolas

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

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

Short answer

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

Step-by-step solution

Step 1. Given information

Equation of an hyperbola is given as $16y^2-x^2=144$.

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,

$\begin{array}{l}16y^2-x^2=144\\ \Rightarrow \frac{16}{144}{y}^2-\frac{1}{144}{x}^2=\frac{144}{144}\\ \Rightarrow \frac{y^2}{9}-\frac{x^2}{144}=1\\ \Rightarrow \frac{y^2}{3^2}-\frac{x^2}{{12}^2}=1\end{array}$

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

$a=3$

$b=12$

Hence the vertices are $(0,\pm 3)$

And the transverse axis is vertical.

Therefore graph labeled as 2 is the correct answer.