Question 1184209
Here's how to find the equation of the hyperbola with the given conditions:

**1. General Equation:**

Since the transverse axis is along the x-axis, the general equation of the hyperbola is:

(x²/a²) - (y²/b²) = 1

where:
* a is the distance from the center to a vertex
* b is related to the distance from the center to a focus (c) by the equation c² = a² + b²

**2. Given Information:**

* Length of transverse axis = 2a (This is consistent with the general equation)
* Vertex bisects the line segment joining the center and focus.

**3. Interpreting the Bisection Condition:**

Let the center of the hyperbola be at (0,0).
Let a vertex be at (a, 0) (since the transverse axis is along the x-axis).
Let a focus be at (c, 0).

The condition states that the vertex bisects the line segment joining the center and the focus.  This means the x-coordinate of the vertex is the midpoint of the x-coordinates of the center and the focus.

(0 + c) / 2 = a
c = 2a

**4. Relating a, b, and c:**

We know that c² = a² + b².  Substitute c = 2a:

(2a)² = a² + b²
4a² = a² + b²
3a² = b²
b = a√3

**5. Final Equation:**

Substitute b = a√3 back into the general equation of the hyperbola:

(x²/a²) - (y²/(3a²)) = 1

We can simplify this further by multiplying the entire equation by 3:

(3x²/a²) - (y²/a²) = 3

Or, multiplying by a^2

3x^2 - y^2 = 3a^2

Therefore, the equation of the hyperbola is (x²/a²) - (y²/(3a²)) = 1 or 3x^2 - y^2 = 3a^2.  Since 'a' can be any positive constant related to the hyperbola's specific dimensions, this is the general form of the equation given the stated conditions.