Answer:
The equation of the hyperbola in standard form is
[tex]\frac{y^{2}}{100}-\frac{4x^{2}}{625}=1[/tex]
Step-by-step explanation:
* We will take about the standard form equation of the hyperbola
- If the given coordinates of the vertices (0 , a) and (0 , -a)
∴ The transverse axis is the y-axis. (because x = 0)
- If the given asymptotes at y = ± (b/a) x
∴ Use the standard form ⇒ y²/a² - x²/b² = 1
* Lets use this to solve our problem
∵ The vertices are (0 , 10) and (0 , -10)
∴ a = ±10
∴ a² = 100
∵ The asymptotes at y = ± 5/4 x
∴ ± 5/4 = ± b/a
∵ a = ± 10
∴ ± 5/4 = ± b/10 ⇒ using cross multiplication
∴± (4b) = ± (5 × 10) = ± 50 ⇒ divide both sides by 4
∴ b = ± 25/2
∴ b² = 625/4
* Now Lets write the equation
* y²/100 - x²/(625/4) = 1
∵ x² ÷ 625/4 = x² × 4/625 = (4x²/625)
∴ y²/100 - 4x²/625 = 1
* The equation of the hyperbola in standard form is
[tex]\frac{y^{2}}{100}-\frac{4x^{2}}{625}=1[/tex]
