HELP with some circles

• Sep 28th 2008, 01:37 PM
pyrosilver
HELP with some circles
Find the center and the radius of the circle x^2 + y^2 - 2ax + 4by = 0.

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Does every equation of the form x^2 + mx + y^2 + ny = p represent a circle? Explain.
• Sep 28th 2008, 01:39 PM
Matt Westwood
Complete the square on x and y and gather terms into something like:

(x+p)^2 + (y+q)^2 = r^2

Then (-p, -q) is the centre and r is the radius.

It's not a circle if r^2 = 0 in which case it's a single point.

Or when r^2 is negative in which case there are no points at all.
• Sep 28th 2008, 01:44 PM
bkarpuz
Quote:

Originally Posted by pyrosilver
Find the center and the radius of the circle x^2 + y^2 - 2ax + 4by = 0.

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Does every equation of the form x^2 + mx + y^2 + ny = p represent a circle? Explain.

Let $A,B,C\in\mathbb{R}$,
then
$x^{2}+y^{2}+Ax+By+C=0$
represents a circle provided that
$A^{2}+B^{2}-4C>0$
and the circle is centered at
$\bigg(-\frac{A}{2},-\frac{B}{2}\bigg)$
$\frac{\sqrt{A^{2}+B^{2}-4C}}{2}.$
Also you may try squaring the given equation to get $(x-m)^{2}+(y-n)^{2}-r^{2}=0$, which is centered at $(m,n)$ with a radius of $r$.