If the vertex of a hyperbola bisects the distance be­tween its centre and the corresponding focus, then the ratio of the square of its conjugate axis to the square of half the distance between the foci is

Options

A. `4//3`
B. `4//sqrt(3)`
C. `2//sqrt(3)`
D. `3//4`

Answer

To solve the problem, we need to find the ratio of the square of the conjugate axis to the square of half the distance between the foci of a hyperbola, given that the vertex bisects the distance between its center and the corresponding focus.

### Step-by-step Solution:

1. **Understanding the Hyperbola**:
The standard form of a hyperbola is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
Here, \(a\) is the distance from the center to the vertex along the transverse axis, and \(b\) is the distance from the center to the vertex along the conjugate axis.

2. **Identifying the Center and Foci**:
For this hyperbola, the center is at the origin (0, 0). The foci are located at \((ae, 0)\) and \((-ae, 0)\), where \(e\) is the eccentricity of the hyperbola. The distance between the foci is \(2ae\).

3. **Finding the Vertex and Midpoint**:
The vertices of the hyperbola are at \((a, 0)\) and \((-a, 0)\). The vertex bisects the distance between the center and the focus. The midpoint \(M\) of the segment connecting the center (0, 0) and one focus \((ae, 0)\) is given by:
\[ M = \left( \frac{0 + ae}{2}, \frac{0 + 0}{2} \right) = \left( \frac{ae}{2}, 0 \right) \]
Since the vertex is at \((a, 0)\), we have:
\[ a = \frac{ae}{2} \]
This simplifies to:
\[ 2a = ae \implies e = 2 \]

4. **Calculating the Conjugate Axis**:
The length of the conjugate axis is given by \(2b\). Therefore, the square of the conjugate axis is:
\[ (2b)^2 = 4b^2 \]

5. **Calculating Half the Distance Between the Foci**:
The distance between the foci is \(2ae\). Thus, half of this distance is:
\[ ae \]
The square of this half distance is:
\[ (ae)^2 \]

6. **Finding the Ratio**:
We need to find the ratio of the square of the conjugate axis to the square of half the distance between the foci:
\[ \text{Ratio} = \frac{4b^2}{(ae)^2} \]

7. **Substituting the Value of \(e\)**:
Since we found \(e = 2\), we substitute this into the equation:
\[ ae = a \cdot 2 = 2a \]
Therefore, the square becomes:
\[ (ae)^2 = (2a)^2 = 4a^2 \]

8. **Final Calculation**:
Now substituting back into the ratio:
\[ \text{Ratio} = \frac{4b^2}{4a^2} = \frac{b^2}{a^2} \]

9. **Using the Eccentricity**:
The eccentricity \(e\) is defined as:
\[ e = \sqrt{1 + \frac{b^2}{a^2}} = 2 \]
Squaring both sides gives:
\[ 4 = 1 + \frac{b^2}{a^2} \implies \frac{b^2}{a^2} = 3 \]

10. **Conclusion**:
Thus, the required ratio of the square of the conjugate axis to the square of half the distance between the foci is:
\[ \text{Ratio} = 3 \]