Identify an equation in standard form for a hyperbola with center (0, 0), vertex (−5, 0), and focus (−6, 0).
in a hyperbola,
a^2 + b^2 = c^2
for yours
(-5)^2 + b^2 = (-6)^2
b^2 = 36-25 = 11
equation:
x^2 / 25 - y^2 / 11 = 1
To find the equation of a hyperbola given its center, vertex, and focus, follow these steps:
1. Identify the coordinates of the center, (h, k), which are given as (0, 0) in this case.
2. Determine the direction in which the hyperbola opens. In this case, since the focus is to the left of the vertex, the hyperbola opens horizontally.
3. Use the formula for the horizontal form of a hyperbola to determine the equation. The equation is given by:
(x - h)^2/a^2 - (y - k)^2/b^2 = 1
Where:
- (h, k) are the coordinates of the center.
- a is the distance from the center to the vertex.
- b is the distance from the center to the foci.
Let's calculate the values of a and b:
- The vertex is at (-5, 0), which means that a = 5 (the absolute value of the x-coordinate of the vertex minus the x-coordinate of the center).
- The focus is at (-6, 0), which means that b = 1 (the absolute value of the x-coordinate of the focus minus the x-coordinate of the center).
Now, substitute the values into the equation:
(x - 0)^2/5^2 - (y - 0)^2/1^2 = 1
Simplifying the equation gives us the standard form of the hyperbola:
x^2/25 - y^2 = 1
Therefore, the equation in standard form for a hyperbola with the given center, vertex, and focus is x^2/25 - y^2 = 1.