Chapter 4 · Geometry and Equations · Concept 3
4-25 Points of Intersection of Two Circles
Solving two circles together sounds much worse than it is. Both equations begin with x² + y², so subtracting one from the other wipes out every square at once and leaves a plain straight line — and that line runs through both crossing points.
Guiding question
Where do the two signal ranges overlap?
Learning outcome. Find the coordinates of the points of intersection of two circles.
Subtract, and the squares disappear
To find where two circles cross, or the line through those points, solve the two equations together — and the way to do that is to subtract.
x² + y² + px + qy + r = 0
x² + y² + lx + my + n = 0
subtract → (p − l)x + (q − m)y + (r − n) = 0
A straight line — and every point shared by both circles is on it.
asked for the LINE
Subtract, tidy, stop. That is the entire answer — one step.
asked for the POINTS
Subtract to get the line, then substitute it into either circle. From here it is lesson 4-20.
Get both into general form first.
The squares only cancel if both equations are written the same way. A circle given as (x − 4)² + (y − 2)² = 16 must be multiplied out to x² + y² − 8x − 4y + 4 = 0 before it can be subtracted from anything. Subtracting a bracketed form from an expanded one leaves a mess and the method looks as though it has failed.
Why the line goes through both points.
A crossing point makes the first equation zero, and it makes the second one zero too. So it makes their difference zero as well — which is exactly what being on that line means. Nothing was assumed; it follows from the two circles alone.
The purple line was not drawn by hand — it came out of subtracting the two equations, and it lands on both crossing points by itself.
x² + y² − 10 = 0
x² + y² + 2x − y − 5 = 0
2. Find where x² + y² = 10 and x² + y² + 2x − y = 5 cross.
- 1
Write both equations out, ready to subtract.
x2 + y2 = 10 …(i) x2 + y2 + 2x − y = 5 …(ii)Both already start with x² + y², so no tidying is needed here.
Work with a partner. Question 2 gives one circle in bracket form. Before subtracting, agree on what it becomes when multiplied out. If you disagree about a sign, expand it a second time slowly — this is where the question is won or lost.
1. Find the line through the crossings.
(b) has no y in it, and that is fine.
The two circles are the same size and sit side by side, so the y² terms and the y terms both cancel. What is left, x = 1/2, is a vertical line halfway between the two centres — exactly where you would expect the crossings to be. An answer with one letter missing is not an incomplete answer; it is a vertical line.
Two omnidirectional radar stations are tracking a drone. Station Alpha covers x² + y² = 25 and Station Beta covers (x + 5)² + (y + 5)² = 65. To pinpoint where the drone crosses between the two perimeters, find the intersection points of their signal boundaries.
Subtracting before both are in general form.
A bracketed circle must be multiplied out first, or the squares refuse to cancel and the method appears broken.
Sign slips while subtracting.
Every term of the second equation changes sign, including the constant. Write the subtraction out in full rather than doing it in your head.
Stopping at the line when points were asked for.
The line is halfway. Substitute it into either circle and solve — and read the question to see which was wanted.
Substituting back into the harder circle.
Both give the same answers, so choose the one with fewer terms. Usually that is the plain x² + y² = r² one.
Checking against only one circle.
A crossing point belongs to both. Test it in both, or the check has proved nothing.
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