Find the vertices, foci, and asymptotes of the hyperbola.
x^2/7 - y^2 = 1
To find the vertices, foci, and asymptotes of the hyperbola given by the equation x^2/7 - y^2 = 1, we can start by rearranging the equation into the standard form of a hyperbola.
The general equation for a hyperbola in standard form is (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 along the x-axis, and b is the distance from the center to a vertex along the y-axis.
Comparing the given equation x^2/7 - y^2 = 1 to the standard form, we see that a^2 = 7 and b^2 = 1. Since a^2 is positive, we know that the hyperbola opens horizontally.
Next, we can find the coordinates of the center. In this case, since there is no term involving x^2, we can infer that the x-coordinate of the center is 0, and since there is a term involving y^2, we can infer that the y-coordinate of the center is also 0. Therefore, the center of the hyperbola is (0, 0).
To find the vertices, we need to add/subtract the value of a (in this case, √7) to/from the x-coordinate of the center. So, the vertices are located at (-√7, 0) and (√7, 0).
The distance from the center to a vertex along the x-axis is a = √7. Similarly, b = 1 represents the distance from the center to a vertex along the y-axis.
Now, let's find the foci of the hyperbola. The foci of a hyperbola can be found using the equation c^2 = a^2 + b^2, where c is the distance from the center to a focus. In this case, a^2 = 7 and b^2 = 1, so c^2 = 7 + 1 = 8. Therefore, c = √8 = 2√2.
The foci are located along the x-axis, so the coordinates of the foci are (-2√2, 0) and (2√2, 0).
Finally, let's find the asymptotes of the hyperbola. The equations of the asymptotes of a hyperbola in standard form are given by y = ±(b/a)(x - h) + k, where (h, k) is the center of the hyperbola.
In this case, the center is (0, 0), a = √7, and b = 1. Therefore, the equations of the asymptotes are y = ±(1/√7)x.
To summarize:
- The center of the hyperbola is (0, 0).
- The vertices are located at (-√7, 0) and (√7, 0).
- The foci are located at (-2√2, 0) and (2√2, 0).
- The equations of the asymptotes are y = ±(1/√7)x.