Find the vertices of the ellipse define $$ \frac{(x+4)^{2}}{4}+\frac{(y-1)^{2}}{25}=1 $$. If necessary, ro

Question

UpStudy AI · Accepted Answer

To find the vertices of the ellipse defined by the equation:

$$
\frac{(x + 4)^2}{4} + \frac{(y - 1)^2}{25} = 1
$$

let's follow these steps:

### Step 1: Identify the Standard Form
The general form of an ellipse centered at $$(h, k)$$ is:

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

where:
- $$a$$ is the semi-major axis (horizontal radius)
- $$b$$ is the semi-minor axis (vertical radius)

### Step 2: Determine the Center and Axes Lengths
Comparing the given equation to the standard form:

$$
\frac{(x + 4)^2}{4} + \frac{(y - 1)^2}{25} = 1
$$

- **Center $$(h, k)$$:** $$(-4, 1)$$
- **$$a^2 = 4$$ ⇒ $$a = 2$$**
- **$$b^2 = 25$$ ⇒ $$b = 5$$**

### Step 3: Determine the Orientation
Since $$b > a$$ ($$5 > 2$$), the major axis is **vertical**.

### Step 4: Calculate the Vertices
For a vertically oriented ellipse, the vertices are located at a distance $$b$$ above and below the center.

- **Upper Vertex:** $$(h, k + b) = (-4, 1 + 5) = (-4, 6)$$
- **Lower Vertex:** $$(h, k - b) = (-4, 1 - 5) = (-4, -4)$$

### Final Answer

The vertices of the ellipse are at the points (–4 , –4) and (–4, 6).
The vertices of the ellipse are at (–4, –4) and (–4, 6).

Find the vertices of the ellipse define \( \frac{(x+4)^{2}}{4}+\frac{(y-1)^{2}}{25}=1 \). If necessary, ro