PREVIOUS ANSWERS ASK YOUR TEACHER Find the center, vertices, foci, and the equations of the asymptotes of the hyperbola. (If an answer does not exist, enter DNE.) $$ \frac{1}{25} y^{2}-\frac{1}{121} x^{2}=1 $$ center $$ \quad(x, y)=(0,0) $$ vertices $$ (x, y)=(0-5) $$ (smaller $$ y $$-value) $$ (x, y)=\left(\begin{array}{r}0,5\end{array}\right) $$ (larger $$ y $$-value) fodi $$ \quad(x, y)=(0,-5 \sqrt{5}) $$ (smaller $$ y $$-value) $$ (x, y)=\left(\begin{array}{|}0,5 \sqrt{5}\end{array}\right) $$ (larger $$ y $$-value) asymptotes $$ y=-\frac{1}{2} x $$ $$ y=\frac{1}{2} x $$ (positive slope)

Question

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**Step 1. Write the Equation in Standard Form**

The given equation is  
$$
\frac{1}{25}y^2 - \frac{1}{121}x^2 = 1.
$$  
This can be rewritten as  
$$
\frac{y^2}{25} - \frac{x^2}{121} = 1.
$$

**Step 2. Identify the Center**

Since the equation is in the form  
$$
\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1,
$$  
the center is at  
$$
(h,k) = (0,0).
$$

**Step 3. Determine the Values of $$ a $$ and $$ b $$**

From the standard form  
$$
\frac{y^2}{25} - \frac{x^2}{121} = 1,
$$  
we identify  
$$
a^2 = 25 \quad \text{and} \quad b^2 = 121.
$$  
Thus,  
$$
a = 5 \quad \text{and} \quad b = 11.
$$

**Step 4. Find the Vertices**

For a hyperbola that opens upward and downward (vertical hyperbola), the vertices are located at  
$$
(0, \pm a).
$$  
So the vertices are  
$$
(0,-5) \quad \text{and} \quad (0,5).
$$

**Step 5. Find the Foci**

For a vertical hyperbola, the foci are given by  
$$
(0, \pm c) \quad \text{where} \quad c^2 = a^2 + b^2.
$$  
Calculate $$ c^2 $$:  
$$
c^2 = 25 + 121 = 146,
$$  
so  
$$
c = \sqrt{146}.
$$  
Thus, the foci are  
$$
(0,-\sqrt{146}) \quad \text{and} \quad (0,\sqrt{146}).
$$

**Step 6. Determine the Equations of the Asymptotes**

For a vertical hyperbola in the form  
$$
\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1,
$$  
the asymptotes are given by  
$$
y = \pm \frac{a}{b}x.
$$  
Substitute $$ a = 5 $$ and $$ b = 11 $$:  
$$
y = \pm \frac{5}{11}x.
$$

**Final Answers**

- Center: $$(0,0)$$
- Vertices: $$(0, -5)$$ and $$(0, 5)$$
- Foci: $$(0, -\sqrt{146})$$ and $$(0, \sqrt{146})$$
- Asymptotes: $$y = \frac{5}{11}x$$ and $$y = -\frac{5}{11}x$$
**Center:** $$(0,0)$$ **Vertices:** $$(0, -5)$$ and $$(0, 5)$$ **Foci:** $$(0, -\sqrt{146})$$ and $$(0, \sqrt{146})$$ **Asymptotes:** $$y = \frac{5}{11}x$$ and $$y = -\frac{5}{11}x$$

PREVIOUS ANSWERS
ASK YOUR TEACHER

Find the center, vertices, foci, and the equations of the asymptotes of the hyperbola. (If an answer does not exist, enter DNE.)

center
vertices (smaller -value)
(larger -value)
fodi (smaller -value)
(larger -value)
asymptotes

(positive slope)

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PREVIOUS ANSWERS ASK YOUR TEACHER Find the center, vertices, foci, and the equations of the asymptotes of the hyperbola. (If an answer does not exist, enter DNE.) center vertices (smaller -value) (larger -value) fodi (smaller -value) (larger -value) asymptotes (positive slope)