AI Tutor
If you swing a ball on a string in a circle, releasing it sends it flying along a straight tangent line. But what path was the string pulling along the entire time?
That inward path lies along the normal line, which is the line perpendicular to the tangent at the exact point of contact.
Does this inward-pointing line always aim directly at the center of the circle, no matter where you draw it?

The Center Property of Normals

In 1637, RenΓ© Descartes used normal lines to analyze curves algebraically, establishing that every normal line to a circle passes through its center.
Here is why: the radius drawn to any point of tangency P(x1,y1)P(x_1, y_1) is naturally perpendicular to the tangent line at that point. Because there is only one line perpendicular to a given line through point PP, the normal line and the radial line segment CPCP must lie on the exact same line.
Because every normal passes through the center, how does this make writing its equation surprisingly simple?

Formulating the Equation

Instead of computing tangent slopes and taking negative reciprocals, you already know two points on the normal line: the contact point P(x1,y1)P(x_1, y_1) and the circle's center C(h,k)C(h, k).
The slope of the normal line mNm_N is simply the slope between C(h,k)C(h, k) and P(x1,y1)P(x_1, y_1):
mN=y1βˆ’kx1βˆ’hm_N = \frac{y_1 - k}{x_1 - h}