Eccentricity of the ellipse `x^2/25+y^2/9=1` is

Options

A. `4/5`
B. `5/4`
C. `3/5`
D. `4/9`

Answer

To find the eccentricity of the ellipse given by the equation \( \frac{x^2}{25} + \frac{y^2}{9} = 1 \), we can follow these steps:

### Step 1: Identify the values of \( a^2 \) and \( b^2 \)
The standard form of an ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
From the given equation \( \frac{x^2}{25} + \frac{y^2}{9} = 1 \), we can identify:
- \( a^2 = 25 \)
- \( b^2 = 9 \)

### Step 2: Determine \( a \) and \( b \)
Next, we find \( a \) and \( b \):
\[ a = \sqrt{25} = 5 \]
\[ b = \sqrt{9} = 3 \]

### Step 3: Check the relationship between \( a \) and \( b \)
Since \( a^2 > b^2 \) (25 > 9), we confirm that the major axis is along the x-axis.

### Step 4: Use the formula for eccentricity
The eccentricity \( e \) of an ellipse is given by the formula:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]

### Step 5: Substitute the values of \( a^2 \) and \( b^2 \)
Substituting the values we found:
\[ e = \sqrt{1 - \frac{9}{25}} \]

### Step 6: Simplify the expression
Calculating the fraction:
\[ 1 - \frac{9}{25} = \frac{25 - 9}{25} = \frac{16}{25} \]
Thus, we have:
\[ e = \sqrt{\frac{16}{25}} \]

### Step 7: Calculate the square root
Taking the square root gives:
\[ e = \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5} \]

### Conclusion
The eccentricity of the ellipse \( \frac{x^2}{25} + \frac{y^2}{9} = 1 \) is \( \frac{4}{5} \).

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