Lines and circles — tangents, chords, and intersections

Where a line meets a circle (substitute, then read the discriminant), and the three circle facts that solve coordinate problems: tangent ⊥ radius, the perpendicular bisector of a chord passes through the centre, and the angle in a semicircle is 90°.

Pure · Coordinate geometry

Lines and circles — tangents, chords, and intersections

Why it works

Where does a line meet a circle? Substitute the line into the circle's equation. You get a quadratic in

x

, and its discriminant tells you the geometry before you even solve it:

$b^2 - 4ac > 0$ — two solutions, the line is a chord (cuts the circle twice);
$b^2 - 4ac = 0$ — one repeated solution, the line is a tangent (touches once);
$b^2 - 4ac < 0$ — no real solutions, the line misses the circle.

This is the way to prove a line is a tangent, or to find the value of a constant that makes it one. Having found

x

, always substitute back into the line (the simpler equation) to get

y

— a coordinate needs both.

Three geometric facts turn up again and again:

Tangent $\perp$ radius. A tangent touches the circle at one point, and at

that point it is perpendicular to the radius drawn to it. So the tangent's gradient is the negative reciprocal of the radius's gradient. This is how you find a tangent's equation at a given point without any calculus.

The perpendicular bisector of a chord passes through the centre. The centre

is equidistant from both ends of any chord, so it lies on that chord's perpendicular bisector. Intersect the perpendicular bisectors of two chords and you've found the centre — the route to the circle through three given points.

The angle in a semicircle is $90^\circ$ . If $AB$ is a diameter and $P$ is

any other point on the circle, then

\angle APB = 90^\circ

. The converse is the useful direction: if

\angle APB = 90^\circ

then

P

lies on the circle with diameter

AB

— test it with gradients (

m_{PA} \times m_{PB} = -1