If the latus rectum of an ellipse is half of its minor axis then e =

Options

A. `(1)/sqrt(2)`
B. `sqrt(3)/(2)`
C. `2/3`
D. `sqrt(3)/(4)`

Answer

To solve the problem, we need to find the eccentricity \( e \) of an ellipse given that the latus rectum is half of its minor axis. Let's break down the solution step by step.

### Step 1: Understand the definitions
The latus rectum \( L \) of an ellipse is given by the formula:
\[ L = \frac{2b^2}{a} \]
where \( a \) is the semi-major axis and \( b \) is the semi-minor axis.

The length of the minor axis is given by:
\[ \text{Minor Axis} = 2b \]

### Step 2: Set up the equation
According to the problem, the latus rectum is half of the minor axis:
\[ \frac{2b^2}{a} = \frac{1}{2} \times 2b \]
This simplifies to:
\[ \frac{2b^2}{a} = b \]

### Step 3: Simplify the equation
We can cancel \( b \) from both sides (assuming \( b \neq 0 \)):
\[ \frac{2b}{a} = 1 \]
This leads to:
\[ 2b = a \]
or
\[ b = \frac{a}{2} \]

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

### Step 5: Substitute \( b \) in the eccentricity formula
Substituting \( b = \frac{a}{2} \) into the eccentricity formula:
\[ e = \sqrt{1 - \frac{\left(\frac{a}{2}\right)^2}{a^2}} = \sqrt{1 - \frac{a^2/4}{a^2}} = \sqrt{1 - \frac{1}{4}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

### Conclusion
Thus, the eccentricity \( e \) is:
\[ e = \frac{\sqrt{3}}{2} \]