Find the vertices, foci, and asymptotes of the hyperbola.
y^2 /9 - x^2/16 =1
To find the vertices, foci, and asymptotes of a hyperbola, we need to rewrite the equation of the hyperbola in a specific form. The standard form of a hyperbola with a horizontal transverse axis is:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
where (h, k) represents the coordinates of the center of the hyperbola, a is the distance from the center to either vertex, and b is the distance from the center to either foci.
Comparing this standard form with the given equation, we can determine the values of h, k, a, and b:
y^2 / 9 - x^2 / 16 = 1
We can rewrite the equation as:
(x - 0)^2 / 16 - (y - 0)^2 / 9 = 1
Comparing coefficients, we can see that h = 0, k = 0, a = 4, and b = 3.
So, the center of the hyperbola is at (0, 0).
The vertices of the hyperbola can be found by adding and subtracting a from the x-coordinate of the center:
Vertex 1: (h + a, k) = (0 + 4, 0) = (4, 0)
Vertex 2: (h - a, k) = (0 - 4, 0) = (-4, 0)
Therefore, the vertices are (4, 0) and (-4, 0).
Now, to find the foci, we can use the formula:
c = sqrt(a^2 + b^2)
where c is the distance from the center to either focus.
Plugging in the values, we get:
c = sqrt(4^2 + 3^2) = sqrt(16 + 9) = sqrt(25) = 5
The foci can be found by adding and subtracting c from the x-coordinate of the center:
Focus 1: (h + c, k) = (0 + 5, 0) = (5, 0)
Focus 2: (h - c, k) = (0 - 5, 0) = (-5, 0)
Therefore, the foci are (5, 0) and (-5, 0).
Finally, we can find the asymptotes of the hyperbola. The equations of the asymptotes for a hyperbola with a horizontal transverse axis are:
y = ± b/a * x
Plugging in the values, we have:
y = ± 3/4 * x
Therefore, the equations of the asymptotes are y = 3/4 * x and y = -3/4 * x.