If the ecentricity of a hyperbola is `sqrt3` then the ecentricity of its conjugate hyperbola is

Options

A. `sqrt2`
B. `sqrt3`
C. `sqrt(3//2)`
D. `2sqrt3`

Answer

To find the eccentricity of the conjugate hyperbola when the eccentricity of the given hyperbola is \( \sqrt{3} \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given eccentricity**:
The eccentricity of the hyperbola is given as \( e = \sqrt{3} \).

2. **Use the relationship between the eccentricities**:
The relationship between the eccentricity of a hyperbola and its conjugate hyperbola is given by the formula:
\[ \frac{1}{e^2} + \frac{1}{e_c^2} = 1 \]
where \( e \) is the eccentricity of the hyperbola and \( e_c \) is the eccentricity of the conjugate hyperbola.

3. **Substitute the known value**:
Substitute \( e = \sqrt{3} \) into the equation:
\[ \frac{1}{(\sqrt{3})^2} + \frac{1}{e_c^2} = 1 \]
This simplifies to:
\[ \frac{1}{3} + \frac{1}{e_c^2} = 1 \]

4. **Isolate the term with \( e_c \)**:
Rearranging the equation gives:
\[ \frac{1}{e_c^2} = 1 - \frac{1}{3} \]
Simplifying the right side:
\[ \frac{1}{e_c^2} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3} \]

5. **Solve for \( e_c^2 \)**:
Taking the reciprocal gives:
\[ e_c^2 = \frac{3}{2} \]

6. **Find \( e_c \)**:
Finally, take the square root to find the eccentricity of the conjugate hyperbola:
\[ e_c = \sqrt{\frac{3}{2}} = \frac{\sqrt{3}}{\sqrt{2}} = \frac{\sqrt{6}}{2} \]

### Final Answer:
The eccentricity of the conjugate hyperbola is \( \frac{\sqrt{6}}{2} \).