Equation of Straight Line Tangent to Circle

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Theorem

Let $\tuple {a, b}$ be the center of a circle $\mathcal C$.

Let $P_n = \tuple {x_n, y_n}$ be any point on $\mathcal C$.


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

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


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

$x = r - a$ for $P = \tuple {r - a, b}$
$x = a - r$ for $P = \tuple {a - r, b}$


Proof

Non-Vertical Tangent Lines

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

$\paren {x - a}^2 + \paren {y - b}^2 = r^2$

where $P = \tuple {a, b}$ 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 with respect to $x$ of both sides of the equation we get:

\(\displaystyle 2 \paren {x - a} + 2 \paren {y - b} \frac {\d y} {\d x}\) \(=\) \(\displaystyle 0\) Derivative of Constant, Chain Rule, Power Rule for Derivatives
\(\displaystyle \leadsto \ \ \) \(\displaystyle \frac {\d y} {\d x}\) \(=\) \(\displaystyle \frac {a - x} {y - b}\)

This is the slope at any point on $\mathcal C$.

From the slope-intercept form of a line, the equation of such a line is:

$y - y_n = m \paren {x - x_n}$

given any point $\paren {x_n, y_n}$ and the gradient $m$.

For $\mathcal T$:

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

Thus the equation of $\mathcal T$ is:

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

$\Box$


Vertical Tangent Lines


Sources