Circle equation is a mathematical formula that describes all points forming a circle on a coordinate plane. Imagine you have a compass and want to draw a circle on coordinate paper. Well, the circle equation tells us which coordinates the compass pencil tip will pass through.
Why is this useful? Because by knowing the circle equation, we can immediately tell where the center of the circle is and what its radius is without having to draw the circle first.
Sometimes we find circle equations that have already been expanded into general form. For example, from the equation (x−1)2+(y−2)2=9, if we expand it:
(x−1)2+(y−2)2=9
x2−2x+1+y2−4y+4=9
x2+y2−2x−4y+5=9
x2+y2−2x−4y−4=0
This last form is called the general form of circle equation:
x2+y2+Dx+Ey+F=0
If we have an equation in general form, we can convert it back to standard form using completing the square technique.
Not all equations of the form x2+y2+Dx+Ey+F=0 are circle equations. The condition is D2+E2−4F>0. If this value is zero, then it's just a single point, and if negative, then there's no curve at all.
