State the vertices, foci, and asymptotes of the hyperbola with the equation.
20x^2-25y^2=100
To find the vertices, we need to first isolate the term with a positive coefficient and divide both sides by 100:
20x^2 - 25y^2 = 100
x^2/(5^2) - y^2/(4^2) = 1
The denominator under x^2 is 5^2, which means a^2 = 5^2, so a = 5. The denominator under y^2 is 4^2, which means b^2 = 4^2, so b = 4. The center of the hyperbola is at (0,0).
The vertices are located along the x-axis, and are given by (+-a,0), where a = 5:
Vertex 1: (-5,0)
Vertex 2: (5,0)
To find the foci, we need to use the formula c^2 = a^2 + b^2, where c is the distance from the center to either focus.
c^2 = a^2 + b^2
c^2 = 5^2 + 4^2
c^2 = 41
c ≈ ± 6.4
The foci are located along the x-axis, and are given by (+-c,0):
Focus 1: (-6.4,0)
Focus 2: (6.4,0)
Finally, we need to find the asymptotes. The slope of each asymptote will be given by b/a, which in this case is 4/5. The equation of the asymptotes is y = ±(4/5)x.
Therefore, the equations of the asymptotes are:
y = (4/5)x
y = -(4/5)x
To determine the vertices, foci, and asymptotes of the hyperbola with the equation 20x^2 - 25y^2 = 100, we need to rearrange the equation into the standard form for a hyperbola.
Step 1: Divide both sides of the equation by 100 to simplify:
x^2/5 - y^2/4 = 1
Step 2: Compare the equation to the standard form:
(x - h)^2/a^2 - (y - k)^2/b^2 = 1
From the given equation, we can determine that h = 0, k = 0, a^2 = 5, and b^2 = 4.
Step 3: Find the values of a, b, c, and determine the vertices, foci, and asymptotes.
The value of a can be found by taking the square root of a^2:
a = sqrt(5)
The value of b can be found by taking the square root of b^2:
b = sqrt(4) = 2
The value of c can be found using the formula c = sqrt(a^2 + b^2):
c = sqrt(5 + 4) = sqrt(9) = 3
So, the vertices and foci can be determined as follows:
Vertices:
The vertices are given by (h ± a, k), which in this case is (0 ± sqrt(5), 0).
Therefore, the vertices are (-sqrt(5), 0) and (sqrt(5), 0).
Foci:
The foci are given by (h ± c, k), which in this case is (0 ± 3, 0).
Therefore, the foci are (-3, 0) and (3, 0).
Asymptotes:
The equations of the asymptotes can be determined using the formula y = ± (b/a)x, which in this case is y = ± (2/sqrt(5))x.
Therefore, the equations of the asymptotes are y = (2/sqrt(5))x and y = -(2/sqrt(5))x.
To summarize, for the hyperbola with the equation 20x^2 - 25y^2 = 100, the vertices are (-sqrt(5), 0) and (sqrt(5), 0), the foci are (-3, 0) and (3, 0), and the equations of the asymptotes are y = (2/sqrt(5))x and y = -(2/sqrt(5))x.