A hyperbola has vertices at (0, ±3) and foci at (0, ±7). Find the equations of the asymptotes.
since the foci lie on the y-axis, we have
y^2/b^2 - x^2/a^2 = 1
a=3
c=7
so b^2 = 40
The asymptotes are
y = ±b/a x = ±√40/3 x
equation of hyperbola is
x^2 / 40 - y^2 / 9 = -1
equations of asymptotes:
9x^2 - 40y^2 = 0
(3x - √40y)(3x + √40y) = 0
y = ± 3/√40 x
To find the equations of the asymptotes of a hyperbola, we can use the formula:
y = ±(b/a) * x
where "a" is the distance from the center to either of the vertices, and "b" is the distance from the center to either of the foci.
In this case, the distance from the center to either of the vertices is 3, so a = 3.
The distance from the center to either of the foci is 7, so b = 7.
Now we can substitute these values into the equation of the asymptotes:
y = ±(7/3) * x
Therefore, the equations of the asymptotes of the hyperbola are y = (7/3) * x and y = -(7/3) * x.