To solve the given problem for the ellipse defined by the equation \( 25x^{2} + 9y^{2} - 150x - 90y + 225 = 0 \), we will follow these steps:
### Step 1: Rearranging the Equation
Start by rearranging the ellipse equation to a standard form. We will group the \( x \) and \( y \) terms together.
\[
25x^{2} - 150x + 9y^{2} - 90y + 225 = 0
\]
### Step 2: Completing the Square
Next, we will complete the square for both the \( x \) and \( y \) terms.
1. For \( x \):
\[
25(x^{2} - 6x) = 25[(x - 3)^{2} - 9] = 25(x - 3)^{2} - 225
\]
2. For \( y \):
\[
9(y^{2} - 10y) = 9[(y - 5)^{2} - 25] = 9(y - 5)^{2} - 225
\]
Putting these back into the equation gives:
\[
25(x - 3)^{2} - 225 + 9(y - 5)^{2} - 225 + 225 = 0
\]
This simplifies to:
\[
25(x - 3)^{2} + 9(y - 5)^{2} - 225 = 0
\]
### Step 3: Standard Form of the Ellipse
Rearranging gives:
\[
25(x - 3)^{2} + 9(y - 5)^{2} = 225
\]
Dividing through by 225:
\[
\frac{(x - 3)^{2}}{9} + \frac{(y - 5)^{2}}{25} = 1
\]
### Step 4: Identifying Parameters
From the standard form \(\frac{(x - h)^{2}}{b^{2}} + \frac{(y - k)^{2}}{a^{2}} = 1\):
- Center \((h, k) = (3, 5)\)
- \(a^{2} = 25 \Rightarrow a = 5\)
- \(b^{2} = 9 \Rightarrow b = 3\)
### Step 5: Finding the Eccentricity
The eccentricity \(e\) is calculated using the formula:
\[
c = \sqrt{a^{2} - b^{2}} = \sqrt{25 - 9} = \sqrt{16} = 4
\]
Thus, the eccentricity \(e\) is:
\[
e = \frac{c}{a} = \frac{4}{5}
\]
### Step 6: Finding Vertices
The vertices are located at:
\[
(3, 5 \pm a) = (3, 5 + 5) \text{ and } (3, 5 - 5) = (3, 10) \text{ and } (3, 0)
\]
### Step 7: Finding Foci
The foci are located at:
\[
(3, 5 \pm c) = (3, 5 + 4) \text{ and } (3, 5 - 4) = (3, 9) \text{ and } (3, 1)
\]
### Step 8: Identifying Axes
- Major axis: Vertical (along the \(y\)-axis)
- Minor axis: Horizontal (along the \(x\)-axis)
### Final Summary
- **Eccentricity \(e\)**: \(\frac{4}{5}\)
- **Center**: \((3, 5)\)
- **Vertices**: \((3, 10)\) and \((3, 0)\)
- **Foci**: \((3, 9)\) and \((3, 1)\)
- **Major Axis**: Vertical (along \(y\)-axis)
- **Minor Axis**: Horizontal (along \(x\)-axis)
