If the eccentricity and length of latus rectum of a hyperbola are `sqrt(13)/3 and 10/3` units respectively, then what is the length of the transverse axis?

Options

A. A. `7/2` unit
B. B. 12 unit
C. C. `15/2` unit
D. D. `15/4` unit

Answer

To solve the problem, we need to find the length of the transverse axis of a hyperbola given its eccentricity and the length of its latus rectum.

### Step-by-Step Solution:

1. **Identify the given values:**
- Eccentricity \( e = \frac{\sqrt{13}}{3} \)
- Length of latus rectum \( L = \frac{10}{3} \)

2. **Recall the formulas:**
- For a hyperbola, the eccentricity is given by:
\[ e = \sqrt{1 + \frac{b^2}{a^2}} \]
- The length of the latus rectum is given by:
\[ L = \frac{2b^2}{a} \]
- The length of the transverse axis is given by:
\[ \text{Length of transverse axis} = 2a \]

3. **Using the length of latus rectum:**
- From the formula for the latus rectum:
\[ \frac{2b^2}{a} = \frac{10}{3} \]
- Rearranging gives:
\[ 2b^2 = \frac{10}{3} a \]
- Therefore:
\[ b^2 = \frac{5}{3} a \]

4. **Substituting \( b^2 \) into the eccentricity formula:**
- Substitute \( b^2 \) into the eccentricity formula:
\[ e = \sqrt{1 + \frac{b^2}{a^2}} = \sqrt{1 + \frac{\frac{5}{3} a}{a^2}} = \sqrt{1 + \frac{5}{3a}} \]
- Squaring both sides gives:
\[ e^2 = 1 + \frac{5}{3a} \]
- Substituting \( e = \frac{\sqrt{13}}{3} \):
\[ \left(\frac{\sqrt{13}}{3}\right)^2 = 1 + \frac{5}{3a} \]
- This simplifies to:
\[ \frac{13}{9} = 1 + \frac{5}{3a} \]

5. **Solving for \( a \):**
- Rearranging gives:
\[ \frac{13}{9} - 1 = \frac{5}{3a} \]
- Converting \( 1 \) to a fraction:
\[ \frac{13}{9} - \frac{9}{9} = \frac{4}{9} \]
- Thus:
\[ \frac{4}{9} = \frac{5}{3a} \]
- Cross-multiplying gives:
\[ 4 \cdot 3a = 5 \cdot 9 \]
- Simplifying:
\[ 12a = 45 \quad \Rightarrow \quad a = \frac{45}{12} = \frac{15}{4} \]

6. **Finding the length of the transverse axis:**
- The length of the transverse axis is:
\[ 2a = 2 \cdot \frac{15}{4} = \frac{30}{4} = \frac{15}{2} \]

### Final Answer:
The length of the transverse axis is \( \frac{15}{2} \) units.

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