Position of a Line Relative to a Circle; the Tangent

A straight line can relate to a circle in exactly three ways: it can cut through the circle at two distinct points (a secant), just graze it at exactly one point (a tangent), or miss it entirely. Algebraically, substituting the line's equation into the circle's equation produces a quadratic in one variable, and the nature of its roots — two real roots, one repeated root, or no real roots — tells you which case you're in. Equivalently, and much faster in practice, you can compare the perpendicular distance $d$ from the centre to the line against the radius $r$: the line is a secant if $d<r$, a tangent if $d=r$, and misses the circle entirely if $d>r$. This distance test is almost always the quickest way to decide.

Equation of the tangent at a known point on the circle. If $P(x_1,y_1)$ is a point that actually lies on the circle $S\equiv x^2+y^2+2gx+2fy+c=0$, the tangent to the circle at $P$ has the compact equation

which is obtained from $S=0$ by the standard 'replace $x^2$ with $xx_1$, $y^2$ with $yy_1$, $x$ with $\tfrac{x+x_1}{2}$, $y$ with $\tfrac{y+y_1}{2}$' substitution. It is worth noticing this is also always perpendicular to the radius $CP$, since a tangent line and the radius to the point of contact meet at right angles.

Tangent of a given slope (no point known yet). For the circle $x^2+y^2=r^2$ centred at the origin, the line $y=mx+c$ (slope $m$) touches the circle exactly when the perpendicular distance from the origin equals $r$, which works out to $c=\pm r\sqrt{1+m^2}$. So the tangents of slope $m$ are

— note there are always two parallel tangents of any given slope, one on each side of the circle. For a circle not centred at the origin, shift coordinates to the centre first, apply this result, then shift back.

Length of a chord. If a line cuts a circle of radius $r$ in two points, and the perpendicular distance from the centre to the line is $d$ (necessarily $d<r$ for the line to be a genuine secant), then half the chord, the radius, and $d$ form a right triangle, giving chord length $=2\sqrt{r^2-d^2}$.

Worked Example 1 (distance test). Is the line $3x+4y=20$ tangent to the circle $x^2+y^2=16$?

The circle is centred at the origin with radius $4$. Distance from the origin to the line is $\dfrac{|3(0)+4(0)-20|}{\sqrt{3^2+4^2}}=\dfrac{20}{5}=4$, exactly equal to the radius. So yes, the line is tangent to the circle.

Worked Example 2 (tangent at a point). Find the tangent to $x^2+y^2-4x-2y-4=0$ at the point $(5,1)$. …