If e is the eccentricity of the ellipse `(x^(2))/(a^(2)) + (y^(2))/(b^(2)) = 1 `
(a `lt ` b ) , then ,

Options

A. `b^(2) = a^(2) ( 1 - e^(2))`
B. `a^(2) = b^(2) ( 1 - e^(2))`
C. `a^(2) = b^(2) ( e^(2) - 1 )`
D. `b^(2) = a^(2) ( e^(2)-1)`

Answer

To find the eccentricity \( e \) of the ellipse given by the equation

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

where \( a < b \), we can follow these steps:

### Step 1: Identify the Major and Minor Axes
Since \( a < b \), we know that:
- \( a \) is the semi-minor axis.
- \( b \) is the semi-major axis.

### Step 2: Use the Formula for Eccentricity
The eccentricity \( e \) of an ellipse is given by the formula:

\[ e = \sqrt{1 - \frac{a^2}{b^2}} \]

### Step 3: Substitute the Values
Substituting the values of \( a \) and \( b \) into the formula, we have:

\[ e = \sqrt{1 - \frac{a^2}{b^2}} \]

### Step 4: Simplify the Expression
To simplify, we can rewrite the expression under the square root:

\[ e = \sqrt{\frac{b^2 - a^2}{b^2}} \]

### Step 5: Final Form of Eccentricity
Thus, we can express \( e \) as:

\[ e = \frac{\sqrt{b^2 - a^2}}{b} \]

### Conclusion
The eccentricity \( e \) of the ellipse is given by:

\[ e = \frac{\sqrt{b^2 - a^2}}{b} \]

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