Definition:Tangential Equation of Circle

Definition

Let $\CC$ be a circle embedded in the Cartesian plane of radius $r$ with its center located at the origin.

Let $\LL$ be a straight line in the plane of $\CC$ whose equation is given by:

$(1): \quad l x + m y + n = 0$

such that $l \ne 0$.

Then the equation:

$\paren {l^2 + m^2} r^2 = n^2$

is known as the tangency equation of $\CC$.

Also see

• Condition of Tangency to Circle whose Center is Origin, which justifies the definition

Sources

• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {III}$. The Circle: $13$. Condition that a straight line should touch a given circle