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
State the vertices, foci, and asymptotes of the hyperbola with the equation.
20x^2-25y^2=100
1 answer