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Find the Equation of an Ellipse?

by Zachary
(CA, USA)











































How do I find the equation of the ellipse with the given information.

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May 15, 2012
Equation of an Ellipse
by: Staff


Question:

by Zachary
(CA, USA)


How do I find the equation of the ellipse with the given information?



Answer:

Hi Zachary,



The general equation for the ellipse will look like this:


x²/a² + y²/b² = 1


The endpoints of the minor axis are (0, ±6).

That means the two end points for the minor axis occur when x = 0, y = +6; and x = 0, y = -6.

Therefore, the minor axis occurs along the “y” axis.

Since that is true, the major axis of the ellipse is horizontal (along the “x” axis).


x²/a² + y²/b² = 1


“±a” are the vertex points associated with the major axis, and “±b” are the vertex points associated with the minor axis.

For your problem

Center at (0, 0)
Vertices of major axis: (a, 0) and (-a, 0)
Vertices of minor axis: (0, b) and (0, -b)


Since you know that a > b, the endpoints given in the problem statement represent b.

b = ±6

therefore,

b² = 6²


The equation for the ellipse is partially complete:

x²/a² + y²/b² = 1

x²/a² + y²/6² = 1


The next step is to solve for “a”.


Foci: (c, 0) and (-c, 0)

Since the distance between the foci = 14, “c” can be calculated.

c = 14/2 (half the distance)

c = 7


Since “b” and “c” are now known, “a” can be calculated

b = ±6

c = 7

c² = a² - b²

7² = a² - 6²

49 = a² - 36

a² = 49 + 36

a² = 85

a = √85 = ±9.2195444572929



the equation for the ellipse is:

x²/85 + y²/36 = 1



>>> the final answer is:

x²/85 + y²/36 = 1



Open the following link to view a graph of Ellipse,


(1) If your browser is Firefox, click the following link to VIEW the graph; or if your browser is Chrome, Internet Explorer, Opera, or Safari (2A) highlight and copy the link, then (2B) paste the link into your browser Address bar & press enter:

Use the Backspace key to return to this page

http://www.solving-math-problems.com/images/ellipse-2012-05-15.png








Thanks for writing.

Staff
www.solving-math-problems.com




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