The length of the transverse axis and the conjugate axis of a hyperbola is 2a units and 2b units repectively. If the length of the latus rectum is 4 units of the conjugate axis is equal to one-third of the distance between the foci, then the eccentricity of the hyperbola is

Options

A. `(6)/(b)`
B. 6b
C. `(5)/(b)`
D. `(b)/(6)`

Answer

To solve the problem step by step, we will use the properties of hyperbolas and the relationships between their axes, foci, and eccentricity.

### Step 1: Understand the given information
- The length of the transverse axis is \(2a\).
- The length of the conjugate axis is \(2b\).
- The length of the latus rectum is given as \(4\) units.
- The length of the conjugate axis is equal to one-third of the distance between the foci.

### Step 2: Write down the formula for the latus rectum
For a hyperbola, the length of the latus rectum \(L\) is given by the formula:
\[ L = \frac{2a^2}{b} \]
According to the problem, this length is equal to \(4\) units:
\[ \frac{2a^2}{b} = 4 \]

### Step 3: Rearranging the equation
From the equation above, we can rearrange it to find \(a^2\):
\[ 2a^2 = 4b \implies a^2 = 2b \]

### Step 4: Relate the conjugate axis to the distance between the foci
The distance between the foci of a hyperbola is given by \(2c\), where \(c = ae\) (with \(e\) being the eccentricity). The problem states that the length of the conjugate axis \(2b\) is equal to one-third of the distance between the foci:
\[ 2b = \frac{1}{3}(2c) \implies 2b = \frac{2ae}{3} \]

### Step 5: Simplifying the equation
From the equation \(2b = \frac{2ae}{3}\), we can simplify it:
\[ b = \frac{ae}{3} \]

### Step 6: Substitute \(a^2\) into the equation for \(b\)
We already found that \(a^2 = 2b\). Now substituting \(b = \frac{ae}{3}\) into this equation:
\[ a^2 = 2\left(\frac{ae}{3}\right) \]
This simplifies to:
\[ a^2 = \frac{2ae}{3} \]

### Step 7: Rearranging to find \(e\)
Rearranging gives:
\[ 3a^2 = 2ae \implies 3a = 2e \implies e = \frac{3a}{2} \]

### Step 8: Substitute \(b\) back into the equation
From \(b = \frac{ae}{3}\), substituting \(e = \frac{3a}{2}\):
\[ b = \frac{a \cdot \frac{3a}{2}}{3} = \frac{a^2}{2} \]

### Step 9: Find the relationship between \(a\) and \(b\)
We know \(a^2 = 2b\) and \(b = \frac{a^2}{2}\), so we can substitute \(b\) back into the equation:
\[ a^2 = 2\left(\frac{a^2}{2}\right) \implies a^2 = a^2 \]
This confirms our relationships are consistent.

### Step 10: Calculate the eccentricity
We can find \(e\) using the relationship \(e = \frac{3a}{2}\) and substituting \(b = \frac{a^2}{2}\):
Using \(b\) in terms of \(a\):
\[ e = \frac{3a}{2} \]
Now we can express \(e\) in terms of \(b\):
Using \(b = \frac{ae}{3}\) and substituting \(e = \frac{3a}{2}\):
\[ b = \frac{a \cdot \frac{3a}{2}}{3} = \frac{a^2}{2} \]
Thus, we can conclude that:
\[ e = \frac{3}{\sqrt{2}} \]

### Final Answer
The eccentricity \(e\) of the hyperbola is:
\[ \frac{3}{\sqrt{2}} \]