Find where a line meets a circle
Substitute a line into a circle, solve the resulting quadratic and use the roots to classify the intersection.
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
After substituting a line into a circle, what equation usually results?
What you will learn
- Solve simultaneous line and circle equations
- Find all coordinates of intersection points
- Distinguish a chord, tangent and non-intersecting line from the real roots
Substitution creates a quadratic
Rearrange the line for x or y, then substitute into the circle equation. Solve the resulting quadratic and use each root to find the matching second coordinate.
Two distinct real roots give two intersection points; the line cuts a chord. One repeated real root gives a tangent. No real roots means the line does not meet the circle.
Use the discriminant to classify without losing coordinates
For Ax² + Bx + C = 0, the discriminant Δ = B² − 4AC is positive for two distinct real roots, zero for a repeated root and negative for no real roots.
A repeated root gives the tangent point when the line is tangent. If two roots are found, substitute both; reporting only x-values does not give the required points.
Pause and explain
What does a positive discriminant show for a line-circle intersection?
Worked example
Find the intersections of x² + y² = 25 and y = x + 1.
Show the worked solution
- Substitute y = x + 1: x² + (x + 1)² = 25.
- Simplify to x² + x − 12 = 0 = (x + 4)(x − 3), so x = −4 or 3.
- Using y = x + 1 gives points (−4, −3) and (3, 4).
Answer The line is a chord; intersections are (−4, −3) and (3, 4).
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
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 intersections of (x − 1)² + (y + 2)² = 25 and y = 1.
Give me a hint
Substitute y = 1 and solve for x.
Compare my reasoning
- (x − 1)² + 3² = 25.
- (x − 1)² = 16, so x = 5 or x = −3.
- The points are (5, 1) and (−3, 1).
Two points: (5, 1) and (−3, 1).
Look for these in your work
- Substitution correct
- Both roots retained
- Coordinates completed
Find where y = x meets x² + y² = 10.
Give me a hint
Substitute y = x, then solve and pair both coordinates.
Compare my reasoning
- x² + x² = 10, so 2x² = 10.
- x = ±√5.
- Since y = x, the points are (√5, √5) and (−√5, −√5).
(√5, √5) and (−√5, −√5).
Look for these in your work
- Quadratic formed
- Both roots found
- Ordered pairs satisfy y = x
Does the line 3x + 4y = 26 meet the circle x² + y² = 25? Justify the classification.
Give me a hint
The circle has centre (0, 0) and radius 5; find the perpendicular distance from its centre to the line.
Compare my reasoning
- The distance is |26|/√(3² + 4²) = 26/5 = 5.2.
- This exceeds the radius 5.
- The line is outside the circle and has no real intersection.
No intersection; the centre-to-line distance 5.2 is greater than the radius 5.
Look for these in your work
- Distance formula applied
- Compared with radius
- No-intersection result justified
Common mistakes
- Substituting into only one squared coordinate
- Forgetting to calculate y for each x-root
- Calling one repeated root two intersection points
- Treating a negative discriminant as a tangent
What can you explain now?
How many distinct intersections occur when the substituted quadratic has discriminant zero?
Compare with the explanation
One; the line is tangent
A zero discriminant means one repeated real root and one contact point.
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Make it stick
How do the quadratic roots and discriminant tell you whether a line cuts, touches or misses a circle?
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.