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I have the following equation for a circumference:

$$9 X^2 + 25 Y^2 - 36 X - 50 Y = 154.$$

So far I only used this general equation: $X^2 + Y^2 + A X + B Y + C = 0$, but now $X^2$ and $Y^2$ are not alone and are being multiplied by an integer.

I would like to know how to find out the center and the radius of this equation

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What is your question? – Clive Newstead Sep 11 '12 at 16:59
Get the center and the radius of this equation – Pacha Sep 11 '12 at 16:59
This is an ellipse, not a circle, so does not have a 'radius'; but it does have major and minor axes. I'll elaborate in an answer. – Clive Newstead Sep 11 '12 at 17:01
you might be right! – Pacha Sep 11 '12 at 17:16
up vote 2 down vote accepted

First of all, you ask for the center and radius, but you have an ellipse and not a circle. You can talk about a center but not really a radius. If what you want is to know the general shape of what you have, here is what you do.

Start by completing the square for both $X$ and $Y$.

This will give you something of the form

$$9(X - H)^2 + 25(Y - K)^2 = S$$

Now, divide both sides by $S$ to get something of the form

$$\frac{(X - H)^2}{A^2} + \frac{(Y - K)^2}{B^2} = 1$$

This is a general form of the equation of an ellipse with center $(H, K)$ (not the most general form, by the way). Then $A$ represents the distance from the center to the edge of the ellipse, if traveling only left or right. And $B$ represents the distance from the center to the edge if traveling up or down. So, the $A$ and $B$ represent something similar to a radius, but not a real radius.

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Your equation is that of an ellipse, not a circle. However, if you substitute $x=3X$ and $y=5Y$, then your equation describes a circle in the $(x,y)$-plane. You can then find the properties (centre, major and minor axes) of the ellipse in the $(X,Y)$-plane by considering the properties of the circle, and applying graph transformations $(x,y) \mapsto (X,Y)$.

For instance, if the centre of the circle in the $(x,y)$-plane is $(a,b)$, then the centre of the ellipse in the $(X,Y)$-plane is $(\frac{a}{3}, \frac{b}{5})$, since $X=\dfrac{x}{3}$ and $Y=\dfrac{y}{5}$. You can do something similar to find the major and minor axes if the ellipse.

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