A hyperbola has a vertical transverse axis of length 10 and

A hyperbola has a vertical transverse axis of length 10 and asymptotes of y=(3/2)x8and y=(3/2)x+4. Find the center of the hyperbola, its focal length, and its eccentricity.

Solution

A hyperbola has a vertical transverse axis of length 10 and asymptotes of y=3/2x-8 and y=-3/2x+4. Find the center of the hyperbola,its focal length, and its eccentricity.

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=10=2a

a=5
a^2=25

slope of asymptotes=3/2=b/a

b=3a/2=15/2
b^2=(15/2)^2=56.25

c^2=a^2+b^2=25+56.25=81.25
c=81.259.01 (focal length)

eccentricity:c/a=9.01/51.802

Equations of asymptotes are straight lines that intersect at the center.
Solve as system of two equations:

y=3x/2-8

y=-3x/2+4

add

y=-4

and again subtract the upper euation we get

x=4

now we can say center(4,-4)

focal lenth=9.01

encentricity=1.802

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