A circle actually starts from a very simple definition. A circle is the shape made by gathering every point that's always the same distance (the radius) from one point (the center). With just this one definition, you can derive the equation of a circle yourself, without ever having to memorize it.
Consider a circle with center (a, b) and radius r. If any point on the circle is (x, y), then the distance from this point to the center must always be r. Applying the distance formula between two points directly gives √((x−a)² + (y−b)²) = r, and squaring both sides to clear the square root gives (x−a)² + (y−b)² = r². This is the equation of a circle in standard form.
Expanding this standard form gives x²−2ax+a²+y²−2by+b²=r², and rearranging leaves only x², y², x, y, and a constant term: x²+y²−2ax−2by+(a²+b²−r²)=0. This form is called the general form, x²+y²+Dx+Ey+F=0. Conversely, if you're given the general form, you can find the center and radius by completing the square (grouping the x and y terms separately and turning each into a squared expression) to convert it back to standard form.
Once you know the equation of a circle, a natural question comes up. How can you tell whether a circle and a line meet, and if so, how many times? There are two ways. The first is to substitute the equation of the line into the equation of the circle to get a quadratic equation in x, then look at the sign of its discriminant D. If D>0, there are two distinct real roots, so they meet at two points (intersecting); if D=0, they meet at exactly one point (tangent); and if D<0, there are no real roots, so they don't meet at all.
The second method is more intuitive: compare the distance from the circle's center to the line with the radius. If the distance is smaller than the radius, the line cuts into the circle and meets it at two points; if the distance exactly equals the radius, the line just grazes the circle and becomes a tangent; and if the distance is larger than the radius, the line passes completely outside the circle and doesn't meet it at all. Remarkably, these two methods — the sign of the discriminant, and comparing distance with radius — always give exactly the same conclusion. It's just a matter of looking at the same situation algebraically versus geometrically.
On our activity page, you can use sliders to change the center and radius and see how the standard form and general form connect, and watch how the relationship between a circle and a line changes at the same time from both the discriminant and the distance comparison.