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Question 515860: Find the formula.
Ellipse with foci at (5, 1) and (-1, 1) and contains a point at (1, 3)
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! Find the formula.
Ellipse with foci at (5, 1) and (-1, 1) and contains a point at (1, 3)
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Gleaned from coordinates of foci, this is an ellipse with a horizontal major axis (x changes, but y does not change) 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 equation:
center: (2,1)
Equation: (x-2)^2/a^2+(y-1)^2/b^2=1
From foci info, c=3
c^2=a^2-b^2=9
b^2=a^2-9
using (x,y) coordinates of given point (1, 3) to solve for a
(x-2)^2/a^2+(y-1)^2/b^2=1
(1-2)^2/a^2+(3-1)^2/(a^2-9)=1
1/a^2+4/(a^2-9)=1
a^2-9+4a^2=a^4-9a^2
a^4-9a^2-a^2-4a^2+9
a^4-14a^2+9=0
solving a with quadratic formula: (let student do this)
a^2=13.3246
b^2=a^2-9=4.3246
..
equation:
(x-2)^2/13.3246+(y-1)^2/4.3246=1
comment: I was hoping to get an exact answer but this is the closest I could get. The answer itself, however, is not as important as the method to get the answer.
See the graph below as a visual check on the answer.
y=±(4.3256-(4.3256/13.3246)(x-2)^2)^.5+1
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