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I am looking for a way to understand the last steps found at this site: http://mathdl.maa.org/mathDL/46/?pa=content&sa=viewDocument&nodeId=196&bodyId=203

The page finishes with showing in the modern style of Apollonius ellipse equation as

$ y^2=x \left( p-\frac{p}{2a}x \right) $

together with the comment: "Can this be written in the standard form of an ellipse? Complete the square and see."

I try to get this to the standard form as in: $ \frac{y^2}{a^2}+\frac{x^2}{b^2}=1 $

But the closest my algebra gets me is: $ \frac{y^2}{xp}+\frac{x}{2a}=1 $

Can anyone show me how they are connected?

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    You need the standard form $(y-y_0)^2/a^2+(x-x_0)^2/b^2=1$ instead. This is the equation of an (axis-aligned) ellipse centered at $(x_0,y_0)$ instead of at the origin.2012-11-01

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$ y^2=x \left( p-\frac{p}{2a}x \right)$

$2a\cdot y^2=p(2ax-x^2)=-p\{(x-a)^2-a^2\}$

$2a\cdot y^2+p(x-a)^2=p\cdot a^2$

$\frac{(x-a)^2}{a^2}+\frac{y^2}{\frac{pa}2}=1$

$\left(\frac{x-a}a\right)^2+\left(\frac y{\sqrt\frac{pa}2}\right)^2=1$

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    Thank you! I thought there was a error in your calculations first therefore i made an edit but i was wrong. (Is there any way for me to cancel my edit?)2012-11-04