If the latusrectum of an ellipse is half of its major axis then e is

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 major axis. Let's break down the solution step by step.

### Step 1: Understand the relationship between latus rectum and major axis
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.

### Step 2: Express the condition given in the problem
According to the problem, the latus rectum is half of the major axis. The length of the major axis is \( 2A \), so half of the major axis is:
\[ \frac{1}{2} \times 2A = A \]
Thus, we can set up the equation:
\[ \frac{2B^2}{A} = A \]

### Step 3: Simplify the equation
To simplify the equation, we can multiply both sides by \( A \):
\[ 2B^2 = A^2 \]
Now, divide both sides by \( A^2 \):
\[ \frac{B^2}{A^2} = \frac{1}{2} \]

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

### Step 5: Conclusion
Thus, the eccentricity \( e \) of the ellipse is:
\[ e = \frac{1}{\sqrt{2}} \]

### Final Answer
The answer is \( \frac{1}{\sqrt{2}} \).

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