Find circle intersections and their common chord
Subtract two circle equations to eliminate squared terms, then solve for the intersection points and common chord.
Section 8 · Coordinate geometry of the circle
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What type of equation usually results when two circle equations are subtracted?
What you will learn
- Eliminate x² and y² by subtracting circle equations
- Find the common chord line of two intersecting circles
- Solve for and check the coordinates of their common points
Subtracting circles produces a straight line
When two circle equations are subtracted, x² and y² cancel. The resulting line through their intersection points is the common chord, also called the radical axis.
Substitute this line into either circle to find the common points. A secant gives two points; a tangent gives a repeated point; a negative discriminant gives no real common point.
Check both coordinates in both equations
After solving for one coordinate, use the line to recover the other. Substitute each ordered pair into both original circle equations.
The common chord is the straight line through the shared points. Its endpoints must satisfy both circles, not just the circle used in the final substitution.
Pause and explain
After finding x = 3 for two circles, what should you do next to find their common points?
Worked example
Find the common chord and intersection points of x² + y² = 25 and (x − 6)² + y² = 25.
Show the worked solution
- Expand the second equation and subtract the first: −12x + 36 = 0, so x = 3.
- Substitute into x² + y² = 25: 9 + y² = 25, so y = ±4.
- The common chord is x = 3, with endpoints (3, 4) and (3, −4).
Answer Common chord x = 3; intersections (3, 4) and (3, −4).
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find the common chord and intersections of x² + y² = 25 and x² + (y − 6)² = 25.
Give me a hint
Subtract the equations to eliminate x² and y².
Compare my reasoning
- Expand the second equation and subtract the first to get −12y + 36 = 0.
- Thus y = 3; substitute into x² + y² = 25 to get x² = 16.
- The common chord is y = 3, with points (4, 3) and (−4, 3).
Common chord y = 3; intersections (4, 3) and (−4, 3).
Look for these in your work
- Squared terms eliminated
- Chord line found
- Both intersections recovered
Find the common chord and intersection points of x² + y² = 25 and (x − 8)² + y² = 25.
Give me a hint
Subtract the equations, then substitute the resulting x-value.
Compare my reasoning
- Subtracting gives −16x + 64 = 0, so x = 4.
- Then 16 + y² = 25, so y = ±3.
- The common chord is x = 4, with endpoints (4, 3) and (4, −3).
Common chord x = 4; intersections (4, 3) and (4, −3).
Look for these in your work
- Chord equation solved
- Both y-values found
- Ordered pairs stated
Find the common chord and endpoints of (x − 1)² + y² = 25 and (x − 5)² + y² = 25.
Give me a hint
Subtract to find the radical axis, then substitute into one circle.
Compare my reasoning
- Subtracting the equations gives −8x + 24 = 0, so x = 3.
- In the first circle, (3 − 1)² + y² = 25, so y² = 21.
- The endpoints are (3, √21) and (3, −√21); the common chord is x = 3.
Common chord x = 3; endpoints (3, √21) and (3, −√21).
Look for these in your work
- Correct line obtained
- Exact square root retained
- Both points satisfy the shared chord
Common mistakes
- Subtracting only some terms and failing to cancel both squares
- Mistaking the common chord line for a circle
- Solving one coordinate but omitting the ordered pairs
- Checking the points in only one original circle
What can you explain now?
What happens to x² + y² when two circle equations are subtracted?
Compare with the explanation
The squared terms cancel
Both equations contain the same quadratic terms, leaving a linear relation.
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Make it stick
Why does subtracting circle equations reveal their common chord, and how do you find the endpoints?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.