The eccentricity of an ellipse i s `sqrt(3)/(2)` its length of latus rectum is

Options

A. `1/2` (length of major axis)
B. `1/3` (length of major axis)
C. `1/4` (length of major axis)
D. `2/3` (length of major axis)

Answer

To find the length of the latus rectum of an ellipse given its eccentricity, we can follow these steps:

### Step 1: Understand the formulas for the ellipse
The standard equation of an ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \(a\) is the semi-major axis and \(b\) is the semi-minor axis. The eccentricity \(e\) of the ellipse is given by:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
The length of the latus rectum \(L\) of the ellipse is given by:
\[ L = \frac{2b^2}{a} \]

### Step 2: Substitute the given eccentricity
We are given that the eccentricity \(e = \frac{\sqrt{3}}{2}\). We can substitute this into the eccentricity formula:
\[ \frac{\sqrt{3}}{2} = \sqrt{1 - \frac{b^2}{a^2}} \]

### Step 3: Square both sides
To eliminate the square root, we square both sides:
\[ \left(\frac{\sqrt{3}}{2}\right)^2 = 1 - \frac{b^2}{a^2} \]
This simplifies to:
\[ \frac{3}{4} = 1 - \frac{b^2}{a^2} \]

### Step 4: Rearrange to find \(\frac{b^2}{a^2}\)
Rearranging the equation gives:
\[ \frac{b^2}{a^2} = 1 - \frac{3}{4} = \frac{1}{4} \]

### Step 5: Find the ratio \(b\) in terms of \(a\)
Taking the square root of both sides, we find:
\[ \frac{b}{a} = \frac{1}{2} \]
Thus, we can express \(b\) as:
\[ b = \frac{a}{2} \]

### Step 6: Substitute \(b\) into the latus rectum formula
Now we substitute \(b\) into the formula for the length of the latus rectum:
\[ L = \frac{2b^2}{a} = \frac{2\left(\frac{a}{2}\right)^2}{a} \]
Calculating \(b^2\):
\[ b^2 = \left(\frac{a}{2}\right)^2 = \frac{a^2}{4} \]
Now substituting:
\[ L = \frac{2 \cdot \frac{a^2}{4}}{a} = \frac{2a^2}{4a} = \frac{a}{2} \]

### Step 7: Final expression for the length of the latus rectum
Thus, the length of the latus rectum is:
\[ L = \frac{a}{2} \]

### Conclusion
The length of the latus rectum of the ellipse is \(\frac{a}{2}\).