If the length of the major axis is n times the minor axis of the ellipse, then ecccentricity is

Options

A. `sqrt(n-1)/(n)`
B. `sqrt(n-1)/(n)^(2)`
C. `sqrt(n^(2)-1)/(n^(2))`
D. `sqrt(n^(2)-1)/(n)`

Answer

To find the eccentricity of an ellipse when the length of the major axis is \( n \) times the length of the minor axis, we can follow these steps:

### Step 1: Define the lengths of the axes
Let the length of the minor axis be \( b \). According to the problem, the length of the major axis \( a \) is given by:
\[ a = n \cdot b \]

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

### Step 3: Substitute \( a \) in the eccentricity formula
We substitute \( a = n \cdot b \) into the eccentricity formula:
\[ e = \sqrt{1 - \frac{b^2}{(n \cdot b)^2}} \]

### Step 4: Simplify the expression
Now, simplify the expression inside the square root:
\[ e = \sqrt{1 - \frac{b^2}{n^2 \cdot b^2}} \]
\[ e = \sqrt{1 - \frac{1}{n^2}} \]

### Step 5: Further simplify the expression
Now we can simplify this further:
\[ e = \sqrt{\frac{n^2 - 1}{n^2}} \]
\[ e = \frac{\sqrt{n^2 - 1}}{n} \]

### Conclusion
Thus, the eccentricity \( e \) of the ellipse is:
\[ e = \frac{\sqrt{n^2 - 1}}{n} \]