SOLUTION: Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (0, ±9); asymptotes: y = ±(3)x

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Question 1176577: Find the standard form of the equation of the hyperbola with the given characteristics.
Vertices: (0, ±9); asymptotes: y = ±(3)x
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
Find the standard form of the equation of the hyperbola with the given characteristics.
Vertices: (0, ±9); asymptotes: y+= ±3x
There are two standard forms:
1. %28x-h%29%5E2%2Fa%5E2-%28y-k%29%5E2%2Fb%5E2=1
2. %28y-k%29%5E2%2Fa%5E2-%28x-h%29%5E2%2Fb%5E2=1

The point-slope form for the equations of the asymptotes is:
y= ±+m%28x-h%29%2Bk
Therefore, the equation, y+}= ±3x tell us that h=0 and k=0

The center is the point (0,0).
the vertices given to be, (0, ±9), tells us that the hyperbola is the vertical transverse type with the equation in item 2
%28y-k%29%5E2%2Fa%5E2-%28x-h%29%5E2%2Fb%5E2=1...substitute h and k
%28y-0%29%5E2%2Fa%5E2-%28x-0%29%5E2%2Fb%5E2=1
y%5E2%2Fa%5E2-x%5E2%2Fb%5E2=1

The point (0, 9) allows us to discover the value of "a":
9%5E2%2Fa%5E2-0%5E2%2Fb%5E2=1
81%2Fa%5E2=1
a%5E2=81
a=9
the asymptotes allows us to discover the value of "b":
y=3x, so b=3
Substitute the value of "a" and “b” into equation:
y%5E2%2F9%5E2-x%5E2%2F3%5E2=1
and your equation in standard form is:
y%5E2%2F81-x%5E2%2F9=1