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Question 1184209: Find the equation of hyperbola if transverse axis of hyperbola is along X-axis and its length is 2a while the vertex of the hyperbola bisects the line segment joining the center and focus?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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.
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