The equation to the hyperbola of given transverse axis whose vertex bisects the distance between the centre and focus, is given by

Options

A. `3x^(2) - y^(2) = 3a^(2)`
B. `x^(2) - 3y^(2) = a^(2)`
C. `x^(2) - y^(2) = 3a^(2)`
D. None of these

Answer

To solve the problem, we need to derive the equation of a hyperbola given that the vertex bisects the distance between the center and the focus. Let's break this down step by step.

### Step 1: Understand the standard form of a hyperbola
The standard form of a hyperbola with a horizontal transverse axis is given by:

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

Here, \( (0, 0) \) is the center, \( (a, 0) \) is the vertex, and \( (ae, 0) \) is the focus, where \( e \) is the eccentricity.

**Hint:** Remember the definitions of the center, vertex, and focus of a hyperbola.

### Step 2: Relate the vertex and focus
According to the problem, the vertex bisects the distance between the center and the focus. The midpoint formula gives us:

\[ \text{Midpoint} = \left(\frac{0 + ae}{2}, \frac{0 + 0}{2}\right) = \left(\frac{ae}{2}, 0\right) \]

Since the vertex is at \( (a, 0) \), we can set these equal:

\[ a = \frac{ae}{2} \]

**Hint:** Set the vertex equal to the midpoint to find a relationship between \( a \) and \( e \).

### Step 3: Solve for \( e \)
From the equation \( a = \frac{ae}{2} \), we can multiply both sides by 2:

\[ 2a = ae \]

Dividing both sides by \( a \) (assuming \( a \neq 0 \)) gives:

\[ e = 2 \]

**Hint:** Use algebraic manipulation to isolate \( e \).

### Step 4: Use the eccentricity formula
The eccentricity \( e \) is related to \( a \) and \( b \) by the formula:

\[ e = \sqrt{1 + \frac{b^2}{a^2}} \]

Substituting \( e = 2 \):

\[ 2 = \sqrt{1 + \frac{b^2}{a^2}} \]

**Hint:** Substitute the value of \( e \) into the eccentricity formula.

### Step 5: Square both sides
Squaring both sides gives:

\[ 4 = 1 + \frac{b^2}{a^2} \]

Subtracting 1 from both sides:

\[ 3 = \frac{b^2}{a^2} \]

Multiplying both sides by \( a^2 \):

\[ b^2 = 3a^2 \]

**Hint:** Rearranging equations can help isolate variables.

### Step 6: Write the equation of the hyperbola
Now we can substitute \( b^2 \) back into the standard form of the hyperbola:

\[ \frac{x^2}{a^2} - \frac{y^2}{3a^2} = 1 \]

This simplifies to:

\[ \frac{x^2}{a^2} - \frac{y^2}{3a^2} = 1 \]

**Hint:** Substitute back to find the final form of the hyperbola.

### Final Equation
The equation of the hyperbola is:

\[ \frac{x^2}{a^2} - \frac{y^2}{3a^2} = 1 \]

### Summary of Steps
1. Write the standard form of the hyperbola.
2. Relate the vertex and focus using the midpoint formula.
3. Solve for eccentricity \( e \).
4. Substitute \( e \) into the eccentricity formula.
5. Square both sides to find \( b^2 \).
6. Write the final equation of the hyperbola.