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Question 196188: what is the general equation of the ellipse which has a foci at (-4,2) and (4,2) and passing through (1,3)
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! what is the general equation of the ellipse which has a foci at (-4,2) and (4,2) and passing through (1,3)
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If you plot those points you will see that the center in at (0,0).
Also. since the foci are each "c" away from the center, c = 4.
Then a^2 + b^2 = 16
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The general form is (x^2/a^2) + (y^2/b^2) = 1
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Since the ellipse passes thru (1,3) substitution gives you:
(1/a^2) + (9/b^2) = 1
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You have two equations in a/b and can solve for a and for b.
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a^2 = 16-b^2
Substitute to get:
(1/(16-b^2)) + (9/b^2) = 1
b^2 + 9(16-b^2) = b^2(16-b^2)
-8b^2 + 144 = 16b^2 - b^4
b^4-24b^2+144 = 0
(b^2-12)^2 = 0
b^2 = 12
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Therefore a^2 = 16-b^2 = 4
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Equation:
(x^2/4) + (y^2/12) = 1
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Cheers,
Stan H.
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