Two circles Cambridge IGCSE Additional Mathematics revision
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In plain words
Two circles can sit apart, touch, overlap, or lie one inside the other. Which it is depends on one comparison: the distance between their centres against their radii.
6 things to know
- Let d be the distance between the centres, and r₁ and r₂ the radii.
- If d is greater than r₁ + r₂, the circles are apart and do not meet.
- If d equals r₁ + r₂, they touch at one point, from the outside.
- If d is between the difference of the radii and their sum, they intersect at two points.
- If d equals the difference of the radii, they touch at one point, one inside the other. If d is less than that, one lies inside the other and they do not meet.
- To find where two circles meet, subtract one equation from the other. The squared terms cancel, leaving the equation of a straight line: the common chord. Then solve that line with either circle.
Worked example
Find the points where the circles x² + y² = 25 and (x − 6)² + y² = 13 meet.
- Expand the second: x² − 12x + 36 + y² = 13.
- Subtract it from the first: 12x − 36 = 12, so x = 4. This is the common chord.
- Put x = 4 into the first circle: 16 + y² = 25, so y = 3 or y = −3.
- The circles meet at (4, 3) and (4, −3).
Tips and tricks
- Compare d with the sum and with the difference of the radii. A quick sketch makes the cases obvious.
- Subtracting the two circle equations always gives a straight line. If squared terms are left, the subtraction has gone wrong.
It lands in your notebook with its questions as flashcards.
Two circles: 5 questions and answers
These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.
Two circles have radii 3 and 4, and their centres are 10 apart. What happens?
10 is greater than 3 + 4.
Two circles have radii 5 and 5, and their centres are 10 apart. What happens?
The distance equals the sum of the radii.
Two circles have radii 5 and 3, and their centres are 6 apart. What happens?
6 is between 5 − 3 and 5 + 3.
What do you get when the equation of one circle is subtracted from another's?
It is the line through the points where they meet: the common chord.
Two circles have radii 7 and 2, and their centres are 5 apart. What happens?
The distance equals the difference of the radii.
Quiz
5 questions
Tap an answer and you’ll see straight away whether it’s right, and why.
Worksheet
2 questions, 7 marks. Write your answers on paper, then check them.
Two circles
Cambridge IGCSE Additional Mathematics 0606 · 7 marks · papermunch.org
Name ______________________________ Date ______________
Two circles have centres (1, 2) and (7, 10), with radii 4 and 6. Determine whether they intersect, touch or do not meet.[3]
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They touch. The distance between the centres is √(6² + 8²) = 10, which equals 4 + 6.
Find the equation of the common chord of the circles x² + y² = 20 and x² + y² − 8x + 12 = 0, and the points where they meet.[4]
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The common chord is x = 4. The circles meet at (4, 2) and (4, −2).
Answers: Two circles
- 1. They touch. The distance between the centres is √(6² + 8²) = 10, which equals 4 + 6.
- 2. The common chord is x = 4. The circles meet at (4, 2) and (4, −2).