Chapter 4 · Geometry and Equations · Concept 3
4-20 Points of Intersection of a Circle and a Line
Push a straight line at a circle and only three things can happen: it slices through at two points, it grazes at one, or it misses. Substituting the line into the circle turns all three into one quadratic — and the discriminant tells you which of the three you have without solving anything.
Guiding question
Where does the road cross the roundabout?
Learning outcome. Calculate the points of intersection between a circle and a straight line.
Substitute first. What you do next depends on the question
Solve the circle and the line together.
Rearrange the line so one letter is alone, put it into the circle, and simplify. You always land on a quadratic.
asked for the COORDINATES
Solve the quadratic, then put each answer back into the line to get its partner.
asked for the NUMBER of points
Do not solve it. Just work out D = b2 − 4ac and read its sign.
| sign of D | D > 0 | D = 0 | D < 0 |
|---|---|---|---|
| how many points | 2 points | 1 point | 0 points |
| what it looks like | cuts straight through | just touches (a tangent) | misses completely |
Put the roots back into the LINE.
The quadratic gives you x. The question wants (x, y) — so each x has to go back in to find its y, and that is a whole extra step people forget. Use the line for it, not the circle: the line gives one clean y per x, while the circle would give you a ± and a decision you do not need to make.
Watch D and the picture change together. The moment the line stops cutting and starts touching is exactly the moment D passes through zero.
x² + y² = 4 and y = −x + 1.20
2x² − 2(1.20)x + (1.20)² − 4 = 0 → D = 26.24
D is positive → 2 points. The quadratic has two different roots, and the line cuts clean through.
Nothing was solved to get that count — only the sign of D was read.
1. Find where the circle (x + 2)² + y² = 10 meets the line y = −2x + 1.
- 1
The line already has y on its own, so put it straight into the circle.
(x + 2)2 + (−2x + 1)2 = 10Wherever a y appeared, −2x + 1 goes in its place. Now only one letter is left.
Work with a partner. Before starting question 2, both of you say out loud whether the question wants coordinates or a count. Then check each other’s working stops in the right place — one of you should be solving, the other should not.
1. Find the coordinates.
(a) and (b) each have only ONE point.
The question says “points”, but both quadratics turn out to be perfect squares with a repeated root. That is not a mistake and it is not a trick — those lines are tangents. Write down the single point and say why there is only one.
In (c), substitute for x instead of y.
The line x + 3y − 1 = 0 rearranges to x = 1 − 3y, and the circle already has (x − 1)² in it — so x − 1 becomes simply −3y and the whole thing collapses to 10y² = 10. Making y the substituted letter instead would work too, but with far more algebra. Look at the shape of the circle before deciding which letter to remove.
2. Find the number of points.
Stopping at x and never finding y.
The question asks for points. "x = ±1" is not a point. Each root must go back into the line to collect its y.
Substituting back into the circle instead of the line.
The circle gives y² and a ± you then have to sort out. The line gives one y straight away. Always use the line.
Solving when only a count was wanted.
If the question says "how many", the sign of D is the complete answer. Solving is time you do not have.
Squaring a bracket piece by piece.
(−2x + 1)² is 4x² − 4x + 1, not 4x² + 1. The middle term is where nearly every wrong quadratic on this page comes from.
Calling a repeated root "no answer".
D = 0 means one point, not none. The line is a tangent, and that single point is a full answer.
Sign In to Keep Going
You've reached the end of the free preview. Sign in — and upgrade to Plus or Pro — to keep going with Mathematics.
Full Access
Unlock all 105 lessons