Question

In Problems 29–36, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility.$y^2-x^2=25$

Short answer

$$\begin{gathered} \mathrm{Center}:(0,0) \\ \mathrm{Transverse}\mathrm{axis}:y-axis \\ \mathrm{Vertices}:(0,\pm 5) \\ \mathrm{Foci}:(0,\pm 5\sqrt{2}) \\ \mathrm{Asymptotes}:y=\pm x \end{gathered}$$

Graph of the equation is:

Step-by-step solution

Step 1. Given information 

Equation of hyperbola, $y^2-x^2=25$

Writing the equation in standard form,$\frac{y^2}{25}-\frac{x^2}{25}=1$

Step 2. Finding the center, transverse axis, vertices, foci, and asymptotes of hyperbola

Comparing the given equation with standard form, $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1,b^2=c^2-a^2$

We get,

$$\begin{gathered} a=5,b=5, \\ c=\pm \sqrt{a^2+b^2} \\ \Rightarrow c=\pm \sqrt{5^2+5^2} \\ \Rightarrow c=\pm \sqrt{25+25} \\ \Rightarrow c=\pm \sqrt{50} \\ \Rightarrow c=\pm 5\sqrt{2} \end{gathered}$$

$$\begin{gathered} \mathrm{Center}:(0,0) \\ \mathrm{Transverse}\mathrm{axis}:y-axis \\ \mathrm{Vertices}:(0,\pm \mathrm{a})=(0,\pm 5) \\ \mathrm{Foci}:(0,\pm \mathrm{c})=(0,\pm 5\sqrt{2}) \\ \mathrm{Asymptotes}:y=\pm \frac{a}{b}x \\ \Rightarrow y=\pm \frac{5}{5}x \\ \Rightarrow y=\pm x \end{gathered}$$

Step 3. Graph of the equation

Graph of the equation is: