To find the equations of the asymptotes for the hyperbola centered at the origin with given vertices and foci, follow these detailed steps:
1. Identify the coordinates:
- Vertex: [tex]\( (0, 40) \)[/tex] (which means the transverse axis is vertical)
- Focus: [tex]\( (0, 41) \)[/tex]
2. Determine the distances from the origin:
- The distance to the vertex ([tex]\(a\)[/tex]) is [tex]\(40\)[/tex], so [tex]\(a = 40\)[/tex].
- The distance to the focus ([tex]\(c\)[/tex]) is [tex]\(41\)[/tex], so [tex]\(c = 41\)[/tex].
3. Use the relationship for hyperbolas to find [tex]\(b\)[/tex]. For a hyperbola, the relationship is:
[tex]\[
c^2 = a^2 + b^2
\][/tex]
Plug in the known values of [tex]\(a\)[/tex] and [tex]\(c\)[/tex]:
[tex]\[
41^2 = 40^2 + b^2
\][/tex]
[tex]\[
1681 = 1600 + b^2
\][/tex]
Solving for [tex]\(b^2\)[/tex]:
[tex]\[
b^2 = 1681 - 1600 = 81
\][/tex]
Taking the square root of both sides, we find:
[tex]\[
b = \sqrt{81} = 9
\][/tex]
4. Determine the equations of the asymptotes:
- For a hyperbola with a vertical transverse axis, the equations of the asymptotes are:
[tex]\[
y = \pm \frac{a}{b} x
\][/tex]
- Substitute the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
y = \pm \frac{40}{9} x
\][/tex]
Therefore, the correct equations of the asymptotes are:
[tex]\[
y = \pm \frac{40}{9} x
\][/tex]
However, none of the options provided directly matches this equation. If the given options are the only choices, there may be a mistake in the question or the provided choices. Based on our calculations, the correct asymptotes should be:
[tex]\[
y = \pm \frac{40}{9} x
\][/tex]
