Factoring and Ellipses: How to Solve for Width?

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The discussion revolves around solving for the width of an ellipse when given a height of 10 units above the origin. The user seeks help in converting an equation into factored form and clarifying the relationship between height and the ellipse's dimensions. There is confusion regarding whether the height refers to the vertical distance from the x-axis to the center of the ellipse. Participants discuss the definitions of the major and minor axes in relation to the width. The conversation emphasizes the need for clarity in understanding the geometric properties of ellipses.
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I'm trying to put this into factored form help me out and help me with ellipses I don't know what to do... Would i have to place the C that i solved into a^2 = 10^2 + 25.4^2 http://s65.photobucket.com/albums/h237/runicrice/?action=view&current=scan.jpg
 
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My work is in the url...
 
If I read this correctly, the question is "What is the width of when the height is 10 units above the origin," which is somewhat of a confusing question.

10 units above the origin is point (0,10) which has x=0, and y=10.

Is the height, h = the vertical distance between the x-axis and the center of the ellipse?

Anyway, is the width considered the major axis or minor axis?

One can solve for either.
 
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