Is the ellipse represented by the equation $$ \frac{(x-5)^{2}}{3}+\frac{(y+1)^{2}}{5}=1 $$ oriented horizontally or vertically? Explain how you know. (4 points) B $$ I $$ $$ \underline{\mathrm{u}} $$

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The ellipse given by the equation

$$
\frac{(x-5)^2}{3} + \frac{(y+1)^2}{5} = 1
$$

is **oriented vertically**.

**Explanation:**

1. **Standard Form Comparison:**
   The standard form of an ellipse is

$$
   \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1
   $$

where:
   - $$(h, k)$$ is the center of the ellipse.
   - $$a$$ is the semi-major axis length.
   - $$b$$ is the semi-minor axis length.

2. **Identifying $$a^2$$ and $$b^2$$:**
   In the given equation:
   - The denominator under the $$x$$-term is $$3$$, so $$a^2 = 3$$.
   - The denominator under the $$y$$-term is $$5$$, so $$b^2 = 5$$.

3. **Determining Orientation:**
   - If $$b^2 > a^2$$, the major axis is along the $$y$$-axis, making the ellipse **vertically oriented**.
   - If $$a^2 > b^2$$, the major axis is along the $$x$$-axis, making the ellipse **horizontally oriented**.

Here, $$5 > 3$$, so $$b^2 > a^2$$, indicating that the major axis is vertical.

4. **Conclusion:**
   Since the denominator under the $$y$$-term is larger, the ellipse stretches more along the $$y$$-axis, confirming it is vertically oriented.
The ellipse is **vertically oriented** because the denominator under the $$y$$-term is larger than that under the $$x$$-term.

Is the ellipse represented by the equation \( \frac{(x-5)^{2}}{3}+\frac{(y+1)^{2}}{5}=1 \) oriented horizontally or vertically? Explain how you know. (4 points) B \( I \) \( \underline{\mathrm{u}} \)

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