If length of the transverse axis of a hyperbola is 8 and its eccentricity is `sqrt(5)//2` then the length of a latus rectum of the hyperbola is

Options

A. 1
B. 2
C. `2sqrt(5)`
D. `8//sqrt(5)`

Answer

To find the length of the latus rectum of the hyperbola given the length of the transverse axis and the eccentricity, we can follow these steps:

### Step 1: Identify the given values
- Length of the transverse axis = 8
- Eccentricity (e) = \(\frac{\sqrt{5}}{2}\)

### Step 2: Relate the transverse axis to 'a'
The length of the transverse axis of a hyperbola is given by \(2a\). Therefore, we can find 'a':
\[ 2a = 8 \implies a = \frac{8}{2} = 4 \]

### Step 3: Use the eccentricity formula
The eccentricity of a hyperbola is given by the formula:
\[ e = \sqrt{1 + \frac{b^2}{a^2}} \]
We can substitute the value of 'e' and 'a' into this formula:
\[ \frac{\sqrt{5}}{2} = \sqrt{1 + \frac{b^2}{4^2}} \]
Squaring both sides:
\[ \left(\frac{\sqrt{5}}{2}\right)^2 = 1 + \frac{b^2}{16} \]
\[ \frac{5}{4} = 1 + \frac{b^2}{16} \]

### Step 4: Solve for \(b^2\)
Rearranging the equation:
\[ \frac{5}{4} - 1 = \frac{b^2}{16} \]
\[ \frac{5}{4} - \frac{4}{4} = \frac{b^2}{16} \]
\[ \frac{1}{4} = \frac{b^2}{16} \]
Multiplying both sides by 16:
\[ b^2 = 4 \]

### Step 5: Calculate the length of the latus rectum
The length of the latus rectum (L) of a hyperbola is given by the formula:
\[ L = \frac{2b^2}{a} \]
Substituting the values of \(b^2\) and \(a\):
\[ L = \frac{2 \times 4}{4} = \frac{8}{4} = 2 \]

### Final Answer
The length of the latus rectum of the hyperbola is \(2\).

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