To find the vertices, foci, and asymptotes of the hyperbola with equation 20x^2 - 25y^2 = 100, we need to first write the equation in standard form:
(x^2)/(5) - (y^2)/(4) = 1
From this equation, we can see that the hyperbola has a horizontal transverse axis, since the x^2 term is positive and is divided by a smaller number than the y^2 term. Therefore, the center of the hyperbola is (0,0).
Using the standard form of the hyperbola, we can find the values of a and b:
a^2 = 5, so a = sqrt(5)
b^2 = 4, so b = 2
Now we can find the vertices, foci, and asymptotes:
Vertices:
The vertices are located at (±a,0), so the vertices are at (-sqrt(5),0) and (sqrt(5),0).
Foci:
The foci are located at (±c,0), where c = sqrt(a^2 + b^2). Plugging in our values of a and b, we get:
c = sqrt(5+4) = sqrt(9) = 3
So the foci are at (-3,0) and (3,0).
Asymptotes:
The equations of the asymptotes are y = ±(b/a)x, so the equations of the asymptotes are:
y = ±(2/sqrt(5))x
Therefore, 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.