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

Explanation

To find the relationship involving 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: Write down the formula for eccentricity
The eccentricity \( e \) of an ellipse is given by the formula:

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

### Step 2: Square both sides
To eliminate the square root, we square both sides of the equation:

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

### Step 3: Rearrange the equation
Now, we can rearrange this equation to isolate \( a^2 \):

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

### Step 4: Multiply both sides by \( b^2 \)
Next, we multiply both sides by \( b^2 \):

\[ a^2 = b^2(1 - e^2) \]

### Conclusion
Thus, we have derived the relationship:

\[ a^2 = b^2(1 - e^2) \]

This is the required relation involving the eccentricity \( e \) of the ellipse.