Condition of Tangency

Condition of Tangency

Goal. Find the condition under which a general line $y=mx+c$ (a line of unknown slope $m$ and intercept $c$, not tied to a specific point) is a tangent to the circle $x^2+y^2=a^2$, and find the point of contact in terms of $m$ and $c$.

Setting up two equations for the same line. Write the given line as

If this line touches the circle at some (unknown) point $(x_1,y_1)$, the tangent equation from Section 6.3.1 gives another equation for the same line:

Matching (I) and (II) as the same line. Two equations represent the same line exactly when their coefficients are proportional:

From the first and third ratios: $x_1=\dfrac{-a^2m}{c}$. From the second and third: $y_1=\dfrac{a^2}{c}$.

Using that $(x_1,y_1)$ lies on the circle. Since $x_1^2+y_1^2=a^2$:

Multiplying through by $\dfrac{c^2}{a^2}$:

A line $y=mx+c$ is a tangent to the circle $x^2+y^2=a^2$ if and only if $c^2=a^2m^2+a^2$, i.e. $c=\pm\sqrt{a^2m^2+a^2}$, and the point of contact is $\left(\dfrac{-a^2m}{c},\ \dfrac{a^2}{c}\right)$.

Since $c$ can be either $+\sqrt{a^2m^2+a^2}$ or $-\sqrt{a^2m^2+a^2}$, there are always two tangent lines with any given slope $m$ — one on each side of the circle:

An equivalent, often faster check. Instead of this algebra, a line is tangent to a circle exactly when the perpendicular distance from the circle's centre to the line equals the circle's radius — this distance-based check is usually the quickest way to test tangency or find an unknown constant in a line/circle problem. …