Worked example 1
Intersect a line and circle
Find where y = 3 meets x² + y² = 25.
- Substitute y = 3
- Solve x² + 9 = 25, so x² = 16.
- Use both roots for x.
Answer: (-4, 3) and (4, 3).
Higher Mathematics
Higher Maths Line and circle intersections
Practise line and circle intersections for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Find and interpret line-circle and circle-circle intersections, including tangency cases.
Before you start
- Solve quadratics.
- Substitute a line equation into another equation.
- Recognise a tangent as one repeated intersection.
Explanation
Substitute the line equation into the circle to form a quadratic in one coordinate. Its roots give zero, one or two intersection points.
For two circles, subtracting the equations often removes both squared terms and produces a line through the intersection points.
Visual support
Substitute the line equation into the circle; the real roots give the intersection points
Method and rules
- Discriminant > 0: two intersections; = 0: tangent; < 0: no real intersections
- Substitute each solved coordinate back into the line.
- For two circles, subtract equations before substituting.
Worked examples
Worked example 2
Method check
What does a zero discriminant mean in a line-circle intersection?
- The quadratic has one repeated root.
So: The line is tangent and there is one repeated intersection.
Watch out
- Taking only the positive square root.
- Reporting coordinate values without pairing them.
- Assuming every line meets a circle twice.
Exam reminder
For line and circle intersections, show the defining equation or formula before simplifying. Check that every final value satisfies the original restrictions, interval or context.
Related Higher Maths topics
Continue with the methods that connect most closely to this topic.