Write the equation of the hyperbola with vertex (±3, 0) and focus (-5, 0).
the center is midway between the two vertices, at (0,0)
So we have
c = 5
a = 3
b = 4
So the equation is
x^2/9 - y^2/16 = 1
To find the equation of a hyperbola with the given information, we can use the standard form equation for a horizontal hyperbola:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
Where (h, k) represents the coordinates of the center, a is the distance from the center to a vertex, and b is the distance from the center to a co-vertex.
1. Given that the vertex is at (±3, 0) and the focus is at (-5, 0), we can determine the distance from the vertex to the focus. Since the hyperbola is horizontal, the distance from the vertex to the focus is a, the horizontal distance.
a = distance from vertex to focus = |-5 - 3| = 8
2. Now, we can substitute the values into the standard equation and solve for b^2:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
(x - 3)^2 / 8^2 - (y - 0)^2 / b^2 = 1
(x - 3)^2 / 64 - (y - 0)^2 / b^2 = 1
3. The remaining value to find is b, the distance from the center to a co-vertex. We can use the formula:
c^2 = a^2 + b^2
Where c is the distance from the center to a focus. We already know a (8) and c (5), so we can solve for b:
5^2 = 8^2 + b^2
25 = 64 + b^2
b^2 = 25 - 64
b^2 = -39
Since b^2 is negative, there is no real solution for b. However, this doesn't affect the equation. We can use the imaginary number i to represent the square root of -39.
4. Finally, we can substitute the values we found into the equation:
(x - 3)^2 / 64 - (y - 0)^2 / (-39) = 1
Simplifying, we have:
(x - 3)^2 / 64 + (y - 0)^2 / 39 = 1
Therefore, the equation of the hyperbola with the given vertex and focus is:
(x - 3)^2 / 64 + (y - 0)^2 / 39 = 1
To find the equation of a hyperbola given the vertex and focus, we will use the following formula:
c^2 = a^2 + b^2
where c is the distance from the center to either focus, and a is the distance from the center to either vertex.
In this case, the vertex is given as (±3, 0), and the focus is given as (-5, 0). Since the focus is to the left of the center (vertex), we will use the negative form of the equation.
Step 1: Find the value of a.
a = 3
Step 2: Find the value of c.
c = 5
Step 3: Use the formula c^2 = a^2 + b^2 to solve for b^2.
(5)^2 = (3)^2 + b^2
25 = 9 + b^2
b^2 = 25 - 9
b^2 = 16
Step 4: Write the equation in the standard form.
The standard form of the equation of a hyperbola is:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
In this case, the center of the hyperbola is at (h, k) = (0, 0), so the equation becomes:
x^2 / 3^2 - y^2 / 4^2 = 1
Simplifying this equation, we have:
(x^2 / 9) - (y^2 / 16) = 1
Therefore, the equation of the hyperbola with vertex (±3, 0) and focus (-5, 0) is:
(x^2 / 9) - (y^2 / 16) = 1