Question: Find the foci of this hyperbola.
$$\frac{y^2}{45} - \frac{x^2}{5} = 1$$
Foci = (0, \pm[?]\sqrt{[ ]})
Simplify your answer completely.
graph shape: a vertical hyperbola centered at the origin, opening up and down; position_hint = center
1. **State the problem:** Find the foci of the hyperbola given by $$\frac{y^2}{45} - \frac{x^2}{5} = 1$$.
2. **Identify the form:** This is a vertical hyperbola centered at the origin with equation $$\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$$ where $$a^2 = 45$$ and $$b^2 = 5$$.
3. **Recall the formula for foci of vertical hyperbola:** The foci are at $$ (0, \pm c) $$ where $$ c^2 = a^2 + b^2 $$.
4. **Calculate $$c^2$$:**
$$
c^2 = a^2 + b^2 = 45 + 5 = 50
$$
5. **Calculate $$c$$:**
$$
c = \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}
$$
6. **Write the coordinates of the foci:**
$$
(0, \pm 5\sqrt{2})
$$
**Final answer:** The foci of the hyperbola are at $$ (0, \pm 5\sqrt{2}) $$.