---
tag: 0C6THFRMZR
kind: definition
name: Tangent line to a circle
msc: [51M04]
defines: 0T7DX8NY1X
uses: [060W5CBZP6, 0MP9BCF6DP]
sources:
  - url: https://en.wikipedia.org/wiki/Tangent_lines_to_circles
    locator: Tangent lines to one circle
  - url: https://en.wikipedia.org/wiki/Tangent_lines_to_circles
    locator: Cartesian equation
---

## Definition

Let $C$ be a [[circle|060W5CBZP6]] in the Euclidean plane $\mathbb{R}^2$, and let $T$ be a point of $C$.
A [[line|0MP9BCF6DP]] $\ell$ is tangent to $C$ at $T$ if $\ell \cap C = \{T\}$. The point $T$ is the point of tangency.

## Remarks

A line meets a circle in two points, in one point or in no point. A line with two common points is a secant line.
For the circle $(x - a)^2 + (y - b)^2 = r^2$ and a point $(x_1, y_1)$ on it, the tangent at this point has the equation $(x - x_1)(x_1 - a) + (y - y_1)(y_1 - b) = 0$.
The tangent at $T$ is the line through $T$ perpendicular to the radius to $T$. See [[a tangent is perpendicular to the radius|031JV5S54D]].
From a point outside the circle there are exactly two tangent lines to the circle.

## Explanations

### L1 human

A tangent line just touches a circle at one point and stays outside the circle. For the unit circle, the line $y = 1$ is tangent at $(0, 1)$. The line $y = 0$ is not tangent: it meets the circle at $(1, 0)$ and at $(-1, 0)$.