Chapter 4 · Geometry and Equations · Concept 3
4-21 Points of Intersection of a Circle and a Line (2)
Now you are told how the circle and the line meet, and something in the equations is missing. Two tools can find it — and which one you should pick is decided entirely by where the unknown is hiding.
Guiding question
How big can the circle get before it hits the road?
Learning outcome. Use the discriminant or the distance to a line to find an unknown constant or radius.
Where the unknown sits decides the tool
If the LINE has the unknown → use the discriminant.
Substitute as usual. The unknown ends up inside the quadratic, so D can be written down and set against zero.
2 points → D > 0 · tangent → D = 0 · no points → D < 0
If the RADIUS is the unknown → compare r with d.
d is the distance from the centre to the line, straight from lesson 4-12:
d = |ax₁ + by₁ + c|√(a² + b²)
And d contains no r, so it comes out as a plain number before the unknown is even mentioned.
| how they meet | cuts at 2 points | touches | misses |
|---|---|---|---|
| r against d | r > d | r = d | r < d |
Both tools always work. One is much shorter.
You could force the discriminant through a radius question — but r² lands inside the constant term of the quadratic and the condition turns into an ugly rearrangement. Choosing the right tool does not change the answer; it changes how long you spend getting there, which in an exam is the same thing.
The line stays put, so the purple distance stays put too. Only the radius changes — and it alone decides which of the three cases you are in.
x² + y² = r² and 2x + y − 5 = 0
r = 1.40 · d = 2.24 (fixed)
r < d — the circle is too small to reach. No points shared.
The purple distance never moved. That is the whole reason this tool beats the discriminant when the unknown is r: d does not depend on r at all.
1. The circle x² + y² = 5 and the line y = 2x − k meet at 2 distinct points. Find k.
- 1
The unknown is in the LINE, so substitute and use the discriminant.
x2 + (2x − k)2 = 5k travels into the quadratic with the rest of the line. That is exactly where D can reach it.
Work with a partner. Read both questions and, before writing anything, decide together which tool each one needs. Say why — “the unknown is in the line” or “the unknown is the radius”. Getting that call right is most of the work.
Question 3 is the only one with ≤ instead of <.
“Have points of intersection” means at least one, and a tangent has one. So the boundary counts and the ends are included. Compare with question 5, which says “2 distinct points” — a tangent is not two points, so there the boundary is excluded. Read the wording, not just the numbers.
A coastal radar station maps its coverage as a circular zone centred at the origin, x² + y² = r². A shipping lane follows 3x + 4y − 10 = 0. Maritime privacy law requires the radar’s signal to stay completely clear of the lane — they must share no points at all. Since a radius is a physical distance (r > 0), find the exact range of r that keeps them separate.
Reaching for the discriminant when the radius is unknown.
r² buries itself in the constant term and the condition turns ugly. If r is what you are looking for, compare r with d.
Forgetting r > 0.
"No intersection" gives r < d, but a radius is a real distance. The answer is 0 < r < d, bounded at both ends.
Missing the ± on a tangent question.
Tangent conditions come from |something| = d, and an absolute value opens two ways. m = ±√3 and m = ±4 both have two answers.
Confusing "meets" with "meets at 2 distinct points".
"Meets" includes the tangent, so the boundary counts and the sign is ≤. "2 distinct points" rules the tangent out, so the boundary is excluded and the sign is strict. The wording decides it.
Using the formula on an untidied line.
y = mx + 4 must become mx − y + 4 = 0 before d can be written down — exactly as in lesson 4-12.
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