24. Foci: $$ (0,0),(4,0) $$; major axis of length 6 25. Center: $$ (1,3) $$; vertex: $$ (-2,3) $$; minor axis of length 4 26. Center: $$ (2,-1) $$; vertex: $$ \left(2, \frac{1}{2}\right) $$; minor axis of length 2

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To solve the problems related to the given foci, centers, vertices, and lengths of the axes, we will derive the equations of the ellipses step by step.

### Problem 24
**Given:**
- Foci: $$ (0,0) $$ and $$ (4,0) $$
- Major axis length: 6

**Step 1: Find the center and distance to foci.**
- The center $$ (h, k) $$ is the midpoint of the foci:
  $$
  h = \frac{0 + 4}{2} = 2, \quad k = \frac{0 + 0}{2} = 0 \quad \Rightarrow \quad (h, k) = (2, 0)
  $$
- The distance $$ c $$ from the center to each focus is:
  $$
  c = \frac{4 - 0}{2} = 2
  $$

**Step 2: Find the semi-major axis $$ a $$.**
- The semi-major axis $$ a $$ is half the length of the major axis:
  $$
  a = \frac{6}{2} = 3
  $$

**Step 3: Find the semi-minor axis $$ b $$ using the relationship $$ c^2 = a^2 - b^2 $$.**
$$
c^2 = a^2 - b^2 \quad \Rightarrow \quad 2^2 = 3^2 - b^2 \quad \Rightarrow \quad 4 = 9 - b^2 \quad \Rightarrow \quad b^2 = 5 \quad \Rightarrow \quad b = \sqrt{5}
$$

**Step 4: Write the equation of the ellipse.**
Since the major axis is horizontal:
$$
\frac{(x - 2)^2}{3^2} + \frac{(y - 0)^2}{(\sqrt{5})^2} = 1 \quad \Rightarrow \quad \frac{(x - 2)^2}{9} + \frac{y^2}{5} = 1
$$

### Problem 25
**Given:**
- Center: $$ (1,3) $$
- Vertex: $$ (-2,3) $$
- Minor axis length: 4

**Step 1: Find the semi-major axis $$ a $$.**
- The distance from the center to the vertex gives us $$ a $$:
$$
a = |1 - (-2)| = 3
$$

**Step 2: Find the semi-minor axis $$ b $$.**
- The semi-minor axis $$ b $$ is half the length of the minor axis:
$$
b = \frac{4}{2} = 2
$$

**Step 3: Write the equation of the ellipse.**
Since the major axis is horizontal:
$$
\frac{(x - 1)^2}{3^2} + \frac{(y - 3)^2}{2^2} = 1 \quad \Rightarrow \quad \frac{(x - 1)^2}{9} + \frac{(y - 3)^2}{4} = 1
$$

### Problem 26
**Given:**
- Center: $$ (2,-1) $$
- Vertex: $$ \left(2, \frac{1}{2}\right) $$
- Minor axis length: 2

**Step 1: Find the semi-major axis $$ a $$.**
- The distance from the center to the vertex gives us $$ a $$:
$$
a = \left| -1 - \frac{1}{2} \right| = \frac{3}{2}
$$

**Step 2: Find the semi-minor axis $$ b $$.**
- The semi-minor axis $$ b $$ is half the length of the minor axis:
$$
b = \frac{2}{2} = 1
$$

**Step 3: Write the equation of the ellipse.**
Since the major axis is vertical:
$$
\frac{(x - 2)^2}{1^2} + \frac{(y + 1)^2}{\left(\frac{3}{2}\right)^2} = 1 \quad \Rightarrow \quad \frac{(x - 2)^2}{1} + \frac{(y + 1)^2}{\frac{9}{4}} = 1
$$

### Summary of Equations
1. **Problem 24:** $$\frac{(x - 2)^2}{9} + \frac{y^2}{5} = 1$$
2. **Problem 25:** $$\frac{(x - 1)^2}{9} + \frac{(y - 3)^2}{4} = 1$$
3. **Problem 26:** $$\frac{(x - 2)^2}{1} + \frac{(y + 1)^2}{\frac{9}{4}} = 1$$
The equations of the ellipses are: 1. For foci at (0,0) and (4,0) with a major axis length of 6: $$ \frac{(x - 2)^2}{9} + \frac{y^2}{5} = 1 $$ 2. For center at (1,3) and a vertex at (-2,3) with a minor axis length of 4: $$ \frac{(x - 1)^2}{9} + \frac{(y - 3)^2}{4} = 1 $$ 3. For center at (2,-1) and a vertex at (2,1/2) with a minor axis length of 2: $$ \frac{(x - 2)^2}{1} + \frac{(y + 1)^2}{\frac{9}{4}} = 1 $$

24. Foci: \( (0,0),(4,0) \); major axis of length 6 25. Center: \( (1,3) \); vertex: \( (-2,3) \); minor axis of length 4 26. Center: \( (2,-1) \); vertex: \( \left(2, \frac{1}{2}\right) \); minor axis of length 2

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