SOLUTION: 9x^2 – 25y^2 + 150y – 450 = 0 Find the center, vertices, foci, and asymptotes of the hyperbola

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Question 470747: 9x^2 – 25y^2 + 150y – 450 = 0
Find the center, vertices, foci, and asymptotes of the hyperbola
Answer by lwsshak3(11628) About Me  (Show Source):
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9x^2 – 25y^2 + 150y – 450 = 0
Find the center, vertices, foci, and asymptotes of the hyperbola
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9x^2 – 25y^2 + 150y – 450 = 0
9x^2 – 25y^2 + 150y = 450
9x^2-25(y^2+6y+9)=450-225=225
9x^2-25(y+9)^2=225
x^2/25-(y+9)^2/9=1
This is a hyperbola with horizontal transverse axis, opens sideways.
Standard form: (x-h)^2/a^2-(y-k)^2/b^2=1
Center: (0,-9)
a^2=25
a=5
length of transverse axis=2a=10
vertices: (0ħ5,-9)
..
b^2=9
b=3
length of conjugate axis=2b=6
..
c^2=a^2+b^2=25+9=34
c=√34
Foci: (0ħ√34,-9)
..
Asymptotes:
slope, m=ħb/a=ħ3/5
y= ħmx+b
b=y-intercept=-9 (at center)
Equation of asymptotes:
y=ħ3x/5-9
see graph below as a visual check on answers.
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y=ħ(9x^2/25-9)^.5-9