SOLUTION: What is the equation of an ellipse with foci at (-6,-3),(0,-3) and a major axis of length 10?

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Question 1084952: What is the equation of an ellipse with foci at (-6,-3),(0,-3) and a major axis of length 10?
Answer by ikleyn(52794) About Me  (Show Source):
You can put this solution on YOUR website!
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The center of the ellipse is at the point (-3,-3).

The major axis is horizontal line y= -3.

The major semi-axis is of the length a= 10%2F2 = 5.

The focal distance 2c is equal to  2c = 0 - (-6) = 6.

So, the eccentricity of the ellipse is 3: c = 3 = sqrt%28a%5E2+-+b%5E2%29.

It implies 9 = a%5E2+-b%5E2 = 5%5E2+-b%5E2;  so,  b^2 = 25 - 9 = 16  and  b = 4.


Thus the minor semi-axis is b= 4.


Then the equation of the ellipse is

%28x-%28-3%29%29%5E2%2F5%5E2 + %28y-%28-3%29%29%5E2%2F4%5E2 = 1,   or

%28x%2B3%29%5E2%2F25 + %28y%2B3%29%5E2%2F16 = 1.

Solved.

Your prerequisite is the lesson
    - Ellipse definition, canonical equation, characteristic points and elements
in this site.

Your sample lessons are
    - Standard equation of an ellipse
    - Identify elements of an ellipse given by its standard equation
    - Find the standard equation of an ellipse given by its elements

    - General equation of an ellipse
    - Transform a general equation of an ellipse to the standard form by completing the square
    - Identify elements of an ellipse given by its general equation
in this site.


Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lesson is the part of this online textbook under the topic
"Conic sections: Ellipses. Definition, major elements and properties. Solved problems".