Answer:
The equation of hyperbola is [tex]\frac{y^2}{10^2}-\frac{x^2}{8^2}=1[/tex].
Step-by-step explanation:
Given information: Vertices at (0, ±10) asymptotes at [tex]y=\pm \frac{5}{4}x[/tex].
Vertices are on the y-axis, so given hyperbola is along the y-axis. The standard form for the hyperbola is
[tex]\frac{(y-k)^2}{b^2}-\frac{(x-h)^2}{a^2}=1[/tex]
where, (h,k) is the center of hyperbola. (0,±b) are vertex and [tex]y=\pm \frac{b}{a}x[/tex] are asymptotes.
Vertices at (0, ±10), it means center of the hyperbola is (0,0). So the standard form for the hyperbola is
[tex]\frac{y^2}{b^2}-\frac{x^2}{a^2}=1[/tex] .... (1)
Vertices of hyperbola:
[tex](0,\pm b)=(0, \pm 10)[/tex]
On comparing we get
[tex]b=10[/tex]
The value of b is 10.
Asymptotes of the hyperbola:
[tex]\pm \frac{b}{a}x=\pm \frac{5}{4}x[/tex]
On comparing both sides, we get
[tex]\frac{b}{a}=\frac{5}{4}[/tex]
Substitute b=10.
[tex]\frac{10}{a}=\frac{5}{4}[/tex]
On cross multiplication,
[tex]40=5a[/tex]
Divide both sides by 5.
[tex]8=a[/tex]
The value of a is 8.
Substitute a=8 and b=10 in equation (1) to find the equation of hyperbola.
[tex]\frac{y^2}{10^2}-\frac{x^2}{8^2}=1[/tex]
Therefore the equation of hyperbola is [tex]\frac{y^2}{10^2}-\frac{x^2}{8^2}=1[/tex].
