Equation of Straight Line Tangent to Circle

Theorem
The equation of a non-vertical tangent line on a circle centered at $(a,b)$ at any point $P = (x_n, y_n)$ is given by


 * $y - y_n = \dfrac {a - x_n}{y_n - b}x - x_n$

Proof
From Equation of a Circle, any circle on the $xy$-plane can be written in the form


 * $(x - a)^2 + (y - b)^2 = r^2$

Where $P = (a,b)$ is the center of the circle, and $r$ is the radius. Note that $a$, $b$, and $c$ are constants. From

Use use the interpretation of the derivative as the Slope of the tangent line. Taking the derivative WRT $x$ of both sides of the equation we get:

This is the slope at any point on the circle. From the Slope-Intercept form of a line we have


 * $y - y_n = \dfrac {\mathrm dy}{\mathrm dx} \Big\vert ^{x = x_n} _{y = y_n}x - x_n$


 * $y - y_n = \dfrac {a - x_n}{y_n - b}x - x_n$