Find the equation in standard form of the hyperbola that satisfies the stated condition.
Foci (0,5) and (0,-5),
asymptotes y = x and y =-x
To find the equation in standard form of a hyperbola with the given foci and asymptotes, we can follow these steps:
Step 1: Determine the center of the hyperbola.
The center of the hyperbola is the midpoint between the foci. In this case, the foci are at (0, 5) and (0, -5), so the center is at (0, 0).
Step 2: Determine the distance between the center and each focus.
The distance between the center and each focus is called the "linear eccentricity" and is denoted by c. In this case, c = 5 (since the foci are located at (0, 5) and (0, -5)).
Step 3: Determine the equation of the asymptotes.
The equation of the asymptotes can be written in the form y = mx + b, where m is the slope of the asymptote and b is the y-intercept. In this case, we have two asymptotes: y = x and y = -x. The slopes are ±1, and the y-intercepts are 0.
Step 4: Determine the distance between the center and each vertex.
The distance between the center and each vertex is called the "semi-major axis" and is denoted by a. In this case, the distance between the center and either asymptote is 1. So, a = 1.
Step 5: Determine the equation in standard form.
The standard form of the equation of a hyperbola with a horizontal transverse axis is:
(x - h)² / a² - (y - k)² / b² = 1
where (h, k) is the center of the hyperbola, a is the semi-major axis, and b is the semi-minor axis.
Since the asymptotes are y = x and y = -x (with slopes ±1), the equation is centered at (0, 0), and the semi-major axis is 1, the equation in standard form of the hyperbola is:
(x - 0)² / 1² - (y - 0)² / b² = 1
Simplifying, we have:
x² - y² / b² = 1
Therefore, the equation in standard form of the hyperbola is x² - y² / b² = 1.