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Question 611809: A hyperbola has a vertical transverse axis of length 16 and asymptotes of y=8/5x+2 and y=-8/5x+2. Find the center of the hyperbola,its focal length, and its eccentricity.
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! A hyperbola has a vertical transverse axis of length 16 and asymptotes of y=8/5x+2 and y=-8/5x+2. Find the center of the hyperbola,its focal length, and its eccentricity.
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Standard form of equation for hyperbola with vertical transverse axis: (y-k)^2/a^2-(x-h)^2/b^2
length of vertical transverse axis=16=2a
a=8
a^2=64
..
slope of asymptotes=8/5=b/a
b=8a/5=64/5
b^2=(64/5)^2=163.84
..
c^2=a^2+b^2=64+163.84=227.84
c=√227.84≈15.09 (focal length)
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eccentricity:c/a=15.09/8≈1.89
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Finding center:
Equations of asymptotes are straight lines that intersect at the center.
Solve as system of two equations:
y=8x/5+2
y=-8x/5+2
..
5y=8x+10
5y=-8x+10
add
10y=20
y=2
..
subtract
0=16x+0
x=0
..
Center: (0,2)
focal length: 15.09
eccentricity:1.89
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