Gather up points that are all the same distance from a center, and you get a circle
A circle is "the set of points at a fixed distance (the radius) from one point (the center)." From this single definition, the equation of a circle follows naturally — and it can also explain how many times a circle and a line can meet.
A circle with center (a,b) and radius r is the set of points (x,y) whose distance from the center is exactly r. Applying the distance formula between two points directly gives √((x−a)²+(y−b)²) = r, and squaring both sides gives the equation of a circle: (x−a)² + (y−b)² = r².
Expanding this equation, it can also be written in the general form x²+y²+Dx+Ey+F=0. Use the sliders to change the center and radius, and see how the two forms connect — and how the relationship between a circle and a line changes.
Center a1
Center b-1
Radius r3
Angle θ of a point on the circle40°
Substituting the equation of a line into the equation of a circle gives a quadratic equation in x. The sign of this quadratic's discriminant (D) tells you how many times the circle and line meet — and it gives exactly the same conclusion as comparing "the distance from the center to the line" with "the radius."