Find the standard form of the equation of the hyperbola with the given characteristics. Foci:(+4,0) or (-4,0)
Asymptotes: y=4x or y=-4x.
To find the standard form of the equation of a hyperbola, we can use the information about its foci and asymptotes.
The standard form of the equation of a hyperbola with foci at (±c,0) and asymptotes in the form y = mx or y = -mx is given by:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
where (h,k) represents the center of the hyperbola, and a and b are the lengths of the semi-major and semi-minor axes, respectively.
1. From the given information, we have the foci at (+4,0) and (-4,0). Since the foci are on the x-axis, the center of the hyperbola is at the origin (h, k) = (0, 0).
2. The distance between the foci and the center of the hyperbola is given by c. In this case, c = 4.
3. The distance between the center of the hyperbola and either of the vertices is given by a. Since a > c, we have a = 4.
4. The slope of the asymptotes is given by m = ±(b/a). In this case, m = ±4/4 = ±1.
Thus, we can write the equation of the asymptotes as y = ±x.
5. To find the value of b, we can use the relation b^2 = a^2 - c^2. Here, b^2 = 4^2 - 4^2 = 16 - 16 = 0. As a result, b = 0.
6. Finally, using the values of a, b, h, and k, we can write the equation of the hyperbola in standard form:
(x - 0)^2 / 4^2 - (y - 0)^2 / 0^2 = 1
Simplifying the equation further, we get:
x^2 / 16 - 0 = 1
Hence, the standard form of the equation of the given hyperbola is:
x^2 / 16 = 1