Where a line meets a circle

7 minAS Level1.3.41.3.5

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can find where a line meets a circle, find the length and midpoint of the chord, and use the discriminant to decide whether a line cuts, touches or misses a circle.

Same trick as ever: substitute

Points where a line and a circle meet satisfy both equations. So:

Rearrange the line to $y = \dots$ (or $x = \dots$ ).
Substitute into the circle. You get a quadratic in one letter.
Solve it, then use the line to get the other coordinate of each point.

Pair each $x$ with its own $y$ . Two points, not four.

Worked example: The ends and length of a chord

The line $y = x + 1$ meets the circle $(x - 2)^2 + (y - 1)^2 = 10$ at $A$ and $B$ . Find the coordinates of $A$ and $B$ , and the length of $AB$ .

Have a go first?

Quick check

CalculateQuestion 1 · Apply

The line $y = 3 - x$ meets the circle $x^2 + y^2 = 29$ at two points. Find their coordinates.

point with larger x: x =

point with larger x: y =

other point: x =

other point: y =

Answer the 2 questions above to continue.

Previous lessonCircle facts that find circles Next lessonGraphs, equations and meeting points

The midpoint of the chord is

M(1, 2)

. The gradient of

CM

-1

and the chord's is

1

: perpendicular, as the symmetry of a circle says it must be.

CalculateQuestion 2 · Stretch

The line $y = x - 2$ cuts the circle $(x - 1)^2 + (y + 2)^2 = 25$ at $P$ and $Q$ . Find the length of $PQ$ . Give an exact surd (such as 2√5) or a decimal to 3 significant figures.

PQ =