1. A hyperbola has vertices (+4,0) and one focus (5,0). What is the standard-form equation of the hyperbola?
2. State the vertices, foci, and asymptotes of the hyperbola with the equation
20x^2 - 25y2^ =100
3. Find an equation that models the path of a satellite if its path is a hyperbola, a = 45,000 km, and c = 71,000 km.
Assume that the center of the hyperbola is the origin and the transverse axis is horizontal.
1. Since the hyperbola has vertices at (+4,0) and one focus at (5,0), we can determine that the center of the hyperbola is at (4,0). The distance from the center to a focus is the value of "c" in the standard form equation of a hyperbola. In this case, c = 1.
The standard form equation of a hyperbola with center (h,k), vertices (h ± a, k), and foci (h ± c, k) is given by:
(x-h)^2/a^2 - (y-k)^2/b^2 = 1
Plugging in the values we have:
Center: (4,0)
Vertices: (4 ± a, 0) = (4 ± 1, 0) = (3,0) and (5,0)
Foci: (4 ± c, 0) = (4 ± 1, 0) = (3,0) and (5,0)
Since the transverse axis is on the x-axis, a = 1. Therefore, the equation of the hyperbola is:
(x-4)^2/1 - y^2/b^2 = 1
(x-4)^2 - y^2/b^2 = 1
2. To find the vertices, foci, and asymptotes of the hyperbola with the equation 20x^2 - 25y^2 = 100, we need to rewrite the equation in standard form. Divide both sides by 100 to get:
x^2/5 - y^2/4 = 1
Comparing this equation to the standard form, we can see that a = √5 and b = √4.
Vertices: (±√5, 0)
Foci: c^2 = a^2 + b^2 = 5 + 4 = 9, so c = 3
Foci: (±3, 0)
The asymptotes of the hyperbola are y = ±(b/a)x, so in this case, they are y = ±(√4/√5)x or y = ±(2/√5)x.
3. The equation of a hyperbola with center at the origin, transverse axis along the x-axis, a = 45,000 km, and c = 71,000 km is given by:
x^2/a^2 - y^2/b^2 = 1
Since the center is at the origin, (h,k) = (0,0), and a = 45,000 km and c = 71,000 km. We can find b using the relationship a^2 = b^2 + c^2:
(45,000)^2 = b^2 + (71,000)^2
2,025,000,000 = b^2 + 5,041,000,000
b^2 = 5,041,000,000 - 2,025,000,000
b^2 = 3,016,000,000
b = √3,016,000,000 ≈ 54,937 km
Therefore, the equation of the hyperbola is:
x^2/(45,000)^2 - y^2/(54,937)^2 = 1