Equations of lines and circles

A curve's equation is a membership test: a point (x, y) is on the curve if and only if its coordinates satisfy the equation. This viewpoint turns geometry into computation, one of the most powerful ideas in mathematics.

For lines, you know the form y = mx + p. There is also the Cartesian form ax + by + c = 0, more general since it also covers vertical lines. To recover the slope of a line given in that form (with b nonzero), solve for y: the coefficient of x that appears is the slope.

For circles, everything flows from the definition: the circle with centre C(a, b) and radius r is the set of points at distance r from C. With the distance formula (Pythagoras!), this gives the equation (x - a)² + (y - b)² = r². Mind the signs: (x - 2)² corresponds to a centre with x-coordinate +2, and the right side is r², not r!

Sometimes a circle hides in expanded form, like x² + y² - 4x + 6y - 3 = 0. To unmask it, complete the squares: group the x terms, the y terms, and reveal (x - a)² and (y - b)². The centre and radius can then be read off directly.

With these equations you can test whether a point is on a circle, inside or outside: compare its distance to the centre with the radius. A true geometric GPS!