Find the equation of ellipse having vertices at `(pm 5, 0) ` and foci `(pm 4, 0)`.

Answer

To find the equation of the ellipse with given vertices and foci, we can follow these steps:

### Step 1: Identify the vertices and foci
The vertices of the ellipse are given as \((\pm 5, 0)\) and the foci are given as \((\pm 4, 0)\).

### Step 2: Determine the values of \(a\) and \(c\)
From the vertices \((\pm 5, 0)\), we can conclude that:
- The distance from the center to the vertices is \(a = 5\).

From the foci \((\pm 4, 0)\), we can conclude that:
- The distance from the center to the foci is \(c = 4\).

### Step 3: Use the relationship between \(a\), \(b\), and \(c\)
For an ellipse, the relationship between \(a\), \(b\), and \(c\) is given by the equation:
\[ c^2 = a^2 - b^2 \]

### Step 4: Substitute the known values into the equation
Substituting the known values of \(a\) and \(c\):
\[ 4^2 = 5^2 - b^2 \]
This simplifies to:
\[ 16 = 25 - b^2 \]

### Step 5: Solve for \(b^2\)
Rearranging the equation gives:
\[ b^2 = 25 - 16 = 9 \]

### Step 6: Find the value of \(b\)
Taking the square root of both sides:
\[ b = \sqrt{9} = 3 \]

### Step 7: Write the equation of the ellipse
Since the major axis is along the x-axis, the standard form of the equation of the ellipse is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Substituting the values of \(a\) and \(b\):
\[ \frac{x^2}{5^2} + \frac{y^2}{3^2} = 1 \]
This simplifies to:
\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]

### Final Answer
The equation of the ellipse is:
\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]