SOLUTION: Find the equation of ellipse when it has vertices at (1, -2) and (7, -2) passing through (3, -1)

Question 536679: Find the equation of ellipse when it has vertices at (1, -2) and (7, -2) passing through (3, -1)
Answer by lwsshak3(11628) (Show Source): You can put this solution on YOUR website!
Find the equation of ellipse when it has vertices at (1, -2) and (7, -2) passing through (3, -1)
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Because given x-coordinates of the vertices change while the y-coordinates do not, this is an ellipse with horizontal major axis. Equation is of the standard form: (x-h)^2/a^2+(y-k)^2/b^2=1, a>b, with (h,k) being the (x,y) coordinates of the center.
For given ellipse:
x-coordinate of center=(7+1)/2=4
y-coordinate of center=-2
center:(4,-2)
..
length of horizontal major axis=7-1=6=2a
a=3
a^2=9
..
Equation:
(x-h)^2/a^2+(y-k)^2/b^2=1
(x-4)^2/9+(y+2)^2/b^2=1
solve for b^2 using coordinates of given point (3,-1)
(3-4)^2/9+(-1+2)^2/b^2=1
(-1)^2/9+(1)^2/b^2=1
1/9+1/b^2=1
1/b^2=1-1/9=8/9
b^2=9/8
..
Equation of ellipse:
(x-4)^2/9+(y+2)^2/(9/8)=1
(x-4)^2/9+8(y+2)^2/9=1
see graph below as a visual check on answers:
..
y=�((9-(x-4)^2)/8)^.5-2

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