Given
color(white)("XXX")3x^2+3y^2-6x+12y=0
Let's start by simplifying by dividing everything by the common factor of 3
color(white)("XXX")x^2+y^2-2x+4y=0
Our target will be to convert this into standard circle form:
color(white)("XXX")(x-color(red)a)^2+(y-color(blue)b)^2=color(green)r^2
for a circle with center (color(red)a,color(blue)b) and radius color(green)r
Regrouping the x and y terms separately
color(white)("XXX")(x^2-2xcolor(white)("XX"))+(y^2+4ycolor(white)("XX"))=0
Completing the square for each
color(white)("XXX")(x^2-2xcolor(cyan)(+1))+(y^2+4ycolor(purple)(+4))=0color(cyan)(+1)color(purple)(+4)
Rewriting as squares in standard circle form
color(white)("XXX")(x-color(red)1)^2+(y-color(blue)(""(-2)))^2=color(green)(""(sqrt(5))^2
This gives us
color(white)("XXX")Center at (color(red)1,color(blue)(-2))
color(white)("XXX")Radius of color(green)(sqrt(5))
To get the intercepts, it is probably easier to work from the earlier equation:
color(white)("XXX")x^2+y^2-2x+4y=0
The Y-intercepts occur when x=0
color(white)("XXX")y^2+4y=0
color(white)("XXX")y(y+4)=0
color(white)("XXX")y=0 or y=--4#
The X-intercepts occur when y=0
color(white)("XXX")x^2-2x=0
color(white)("XXX")x(x-2)=0
color(white)("XXX")x=0 or x=2
![enter image source here]()
