The length of latus rectum and the length of conjugate axis of a hyperbola are `4sqrt(3)` and `2sqrt(3)` respectively. Find the length of semi transverse axis

Answer

To solve the problem, we need to find the length of the semi-transverse axis of a hyperbola given the lengths of the latus rectum and the conjugate axis.

### Step-by-step Solution:

1. **Identify the given values:**
- Length of the latus rectum (L) = \(4\sqrt{3}\)
- Length of the conjugate axis (C) = \(2\sqrt{3}\)

2. **Use the formula for the length of the latus rectum:**
The formula for the length of the latus rectum of a hyperbola is given by:
\[ L = \frac{2b^2}{a} \]
Substituting the value of L:
\[ \frac{2b^2}{a} = 4\sqrt{3} \]

3. **Use the formula for the length of the conjugate axis:**
The formula for the length of the conjugate axis is:
\[ C = 2b \]
Substituting the value of C:
\[ 2b = 2\sqrt{3} \]
Dividing both sides by 2 gives:
\[ b = \sqrt{3} \]

4. **Substitute the value of b into the latus rectum formula:**
Now substitute \(b = \sqrt{3}\) into the latus rectum formula:
\[ \frac{2(\sqrt{3})^2}{a} = 4\sqrt{3} \]
Simplifying \((\sqrt{3})^2\) gives us:
\[ \frac{2 \cdot 3}{a} = 4\sqrt{3} \]
This simplifies to:
\[ \frac{6}{a} = 4\sqrt{3} \]

5. **Solve for a:**
Cross-multiplying gives:
\[ 6 = 4\sqrt{3} \cdot a \]
Dividing both sides by \(4\sqrt{3}\) gives:
\[ a = \frac{6}{4\sqrt{3}} = \frac{3}{2\sqrt{3}} \]

6. **Rationalize the denominator:**
To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{3}\):
\[ a = \frac{3\sqrt{3}}{2 \cdot 3} = \frac{\sqrt{3}}{2} \]

7. **Conclusion:**
The length of the semi-transverse axis \(a\) is:
\[ a = \frac{\sqrt{3}}{2} \]

### Final Answer:
Therefore, the length of the semi-transverse axis is \(\frac{\sqrt{3}}{2}\).