SOLUTION: Write the equation of a hyperbola with the gicen characteristics. vertices (-5,3) and (-1,3), foci (-3-2 sqrt5,3) and (-3+2 sqrt5,3)

Question 455752: Write the equation of a hyperbola with the gicen characteristics.
vertices (-5,3) and (-1,3), foci (-3-2 sqrt5,3) and (-3+2 sqrt5,3)
Answer by lwsshak3(11628) (Show Source): You can put this solution on YOUR website!
Write the equation of a hyperbola with the gicen characteristics.
vertices (-5,3) and (-1,3), foci (-3-2 sqrt5,3) and (-3+2 sqrt5,3)
..
Standard form of a hyperbola with horizontal transverse axis: (x-h)^2/a^2-(y-k)^2/b^2=1, with (h,k) being the (x,y) coordinates of the center.
Standard form of a hyperbola with vertical transverse axis: (y-k)^2/a^2-(x-h)^2/b^2=1, with (h,k) being the (x,y) coordinates of the center.
The difference between the two forms is the interchange of (x-h) and (y-k)
..
Since the y-coordinates of the vertices and foci are the same at 3, given hyperbola has a horizontal transverse axis. This is also the y-coordinate of the center. The x-coordinate of the center is the midpoint of the vertices on the transverse axis(-5+(-1)/2=-6/2=-3)
center at (-3,3)
length of the transverse axis=4=2a
a=2
a^2=4
c=distance from center to foci=2√5=4.47..
c^2=a^2+b^2
b^2=c^2-a^2=(2√5)^2-4=20-4=16
b=4
Equation: (x+3)^2/4-(y-3)^2/16=1 (ans)
Asymptotes:y=2x+9, y=-2x-3
see graph below as a visual check on the algebra above.
..
y=(4(x+3)^2-16)^.5+3

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