Equation of Line Tangent to Circle

Theorem

Let $\left({a, b}\right)$ be the center of a circle $\mathcal C$.

Let $P_n = \left({x_n, y_n}\right)$ be any point on $\mathcal C$.

The equation of a non-vertical tangent line $\mathcal T$ to $\mathcal C$ is given by:

$y - y_n = \dfrac {a - x_n}{y_n - b} \left({x - x_n}\right)$

The equations of the vertical tangent lines to $\mathcal C$ are:

$x = r - a$ for $P = \left({r-a, \ b}\right)$

$x = a - r$ for $P = \left({a-r, \ b}\right)$

Proof

Non-Vertical Tangent Lines

From Equation of a Circle, $\mathcal C$ can be described on the $xy$-plane in the form:

$\left({x - a}\right)^2 + \left({y - b}\right)^2 = r^2$

where $P = \left({a, b}\right)$ is the center of the circle and $r$ is the radius.

We use the definition of the derivative as the gradient of the tangent line $\mathcal T$.

Taking the derivative WRT $x$ of both sides of the equation we get:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle 2 \left({x - a}\right) + 2 \left({y - b}\right) \frac {\mathrm dy} {\mathrm dx}\)	\(=\)	\(\displaystyle 0\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Derivative of Constant, Chain Rule, Power Rule
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle \frac {\mathrm dy}{\mathrm dx}\)	\(=\)	\(\displaystyle \frac {a - x}{y - b}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

This is the slope at any point on the circle.

From the gradient-intercept form of a line, given any point $\left({x_n, y_n}\right)$ and the gradient $m$, the equation of such a line is:

$y - y_n = m \left({x - x_n}\right)$

For $\mathcal T$:

$m = \left.{\dfrac {\mathrm dy}{\mathrm dx}}\right\vert^{x = x_n} _{y = y_n}$

Thus the equation of $\mathcal T$ is:

$y - y_n = \dfrac {a - x_n}{y_n - b} \left({x - x_n}\right)$

$\Box$

Sources

• For a video presentation of the contents of this page, visit the Khan Academy.
• Weisstein, Eric W. "Slope-Intercept Form." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Slope-InterceptForm.html