Find equation of an ellipse having vetices `(0, pm5)` and foci `(0,pm4)`.

Answer

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

### Step 1: Identify the values of 'a' and 'b'
The vertices of the ellipse are given as (0, ±5), which means that the distance from the center to the vertices (b) is 5. Thus, we have:
- \( b = 5 \)

The foci are given as (0, ±4), which means that the distance from the center to the foci (c) is 4. Thus, we have:
- \( c = 4 \)

### Step 2: Use the relationship between a, b, and c
For an ellipse, the relationship between 'a', 'b', and 'c' is given by:
\[ c^2 = b^2 - a^2 \]
Since we know the values of 'b' and 'c', we can substitute them into the equation:
\[ 4^2 = 5^2 - a^2 \]
This simplifies to:
\[ 16 = 25 - a^2 \]

### Step 3: Solve for 'a^2'
Rearranging the equation gives:
\[ a^2 = 25 - 16 \]
\[ a^2 = 9 \]

### Step 4: Write the equation of the ellipse
The standard form of the equation of an ellipse centered at the origin with a vertical major axis is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Substituting the values of \( a^2 \) and \( b^2 \):
\[ \frac{x^2}{9} + \frac{y^2}{25} = 1 \]

### Step 5: Rearranging the equation
To express it in a different form, we can multiply through by the least common multiple (LCM) of the denominators:
\[ 25x^2 + 9y^2 = 225 \]

### Final Answer
Thus, the equation of the ellipse is:
\[ 25x^2 + 9y^2 = 225 \]

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