The equation of the hyperbola in standard form is [tex]\frac{(y-4)^{2}}{100}-\frac{(x-1)^{2}}{64}=1[/tex]
Step-by-step explanation:
Let us revise the equation of the hyperbola
The standard form of the equation of a hyperbola with center (h , k) and transverse axis parallel to the y-axis is [tex]\frac{(y-k)^{2}}{a^{2}}-\frac{(x-h)^{2}}{b^{2}}=1[/tex]
∵ The center of the hyperbola is (1 , 4)
∴ h = 1 and k = 4
∵ The transverse axis is parallel to the y-axis
∴ The form of the equation is [tex]\frac{(y-k)^{2}}{a^{2}}-\frac{(x-h)^{2}}{b^{2}}=1[/tex]
∴ The length of the transverse axis is 2a
∵ The length of the transverse axis is 20 units
- Equate 2a by 20 to find a
∴ 2a = 20
- Divide both sides by 2
∴ a = 10
∵ The length of the conjugate axis is 2b
∵ The length of the conjugate axis is 16
- Equate 2b by 16 to find b
∴ 2b = 16
- Divide both sides by 2
∴ b = 8
- Substitute the values of h, k, a, and b in the form of the equation
∴ [tex]\frac{(y-4)^{2}}{(10)^{2}}-\frac{(x-1)^{2}}{(8)^{2}}=1[/tex]
∴ [tex]\frac{(y-4)^{2}}{100}-\frac{(x-1)^{2}}{64}=1[/tex]
The equation of the hyperbola in standard form is [tex]\frac{(y-4)^{2}}{100}-\frac{(x-1)^{2}}{64}=1[/tex]
Learn more:
You can learn more about the hyperbola in brainly.com/question/4054269
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