Tangent to a Circle

Tangent to a Circle

Definition. When a line meets a circle at two coincident points (rather than two distinct points, or none), it is called a tangent to the circle, and the single point where they meet is the point of contact.

Deriving the tangent at a point on the standard circle. Let the circle be $x^2+y^2=r^2$, centred at $O(0,0)$, and let $P(x_1,y_1)$ be a point on it. The slope of the radius $OP$ is $\dfrac{y_1}{x_1}$ (for $x_1\neq0$). A key geometric fact about circles is that the tangent at any point is always perpendicular to the radius drawn to that point. So the tangent's slope $m$ satisfies

(the negative reciprocal of the radius's slope). Using the point-slope form of a line through $P(x_1,y_1)$ with this slope:

Multiplying through by $y_1$ and rearranging:

Since $P(x_1,y_1)$ lies on the circle, $x_1^2+y_1^2=r^2$, so this simplifies to

The equation of the tangent to the circle $x^2+y^2=r^2$ at the point $(x_1,y_1)$ on it is $xx_1+yy_1=r^2$.

Extending to the general-form circle. Following the same method for $x^2+y^2+2gx+2fy+c=0$, the tangent at a point $(x_1,y_1)$ on this circle works out to

A handy way to remember this: starting from the circle's own equation, replace $x^2$ by $xx_1$, $2x$ by $(x+x_1)$, $y^2$ by $yy_1$, and $2y$ by $(y+y_1)$ — a substitution rule that works for the tangent at any point on a circle written in this general form.

Tangent in parametric form. If the point of contact is written using the circle's own parameter as $(r\cos\theta_1, r\sin\theta_1)$, substituting $x_1=r\cos\theta_1$, $y_1=r\sin\theta_1$ into $xx_1+yy_1=r^2$ gives

Solved Example 2 — show $3x-4y+15=0$ is a tangent to $x^2+y^2=9$; find the point of contact

Step 1 — substitute the line into the circle. From $3x-4y+15=0$, $y=\dfrac{3x+15}{4}$. Substituting into $x^2+y^2=9$:

Multiplying through by $16$: $16x^2+9x^2+90x+225=144$, i.e.

Step 2 — check the discriminant. This factors as $(5x+9)^2=0$ — a repeated root, $x=-\dfrac95$. A line meeting a circle in a repeated root meets it at only one (coincident) point, which is exactly the definition of a tangent — so the line is indeed tangent to the circle. …