Chapter 4 · Geometry and Equations · Concept 3
4-24 Positional Relationship Between Two Circles
Two circles can sit five different ways relative to each other, and which one you have is decided by a single number: how far apart their centres are. Compare that distance against the sum and the difference of the radii, and the picture is settled.
Guiding question
Do the two signal ranges overlap, touch, or miss?
Learning outcome. Determine the relative position of two circles — separate, tangent, or intersecting.
One distance, two markers, five answers
Let r be the bigger radius, r′ the smaller, and d the distance between the two centres. Then everything depends on where d falls:
r + r′ < d
are external to each other
too far apart to touch at all
r + r′ = d
touch externally
they just meet, outside to outside
r − r′ < d < r + r′
intersect at 2 points
they overlap and cross twice
r − r′ = d
touch internally
the small one just kisses the inside of the big one
r − r′ > d
one is inside the other
the small one floats inside, never touching
r must be the bigger one.
The whole table is written assuming r > r′. Swap them and r − r′ comes out negative, which is smaller than any distance, so the two internal cases quietly stop working. Sort the radii before you compare anything.
Keep everything exact.
d, r and r′ are nearly always surds. √20 is 2√5, and then 3√5 − √5 = 2√5 matches exactly. Rounding to 4.47 and 2.24 would only ever suggest that they match — and in the touching cases, “suggests” is not an answer.
Start with the small circle buried inside and drag it away. The five cases arrive one after another, always in the same order — because they are five stretches of one number line.
intersect at 2 points
d = 2.40 · r − r′ = 1.8 · r + r′ = 4.2 → r − r′ < d < r + r′
The five cases are just five regions of one number line, split at r − r′ and r + r′.
2. Find the circle with centre (3, 1) that touches (x − 1)² + (y + 3)² = 5 internally.
- 1
Read everything you are given off the second circle.
centre (1, −3), r′ = √5(y + 3)² means the centre has y = −3. The right-hand side is 5, so the radius is √5, not 5.
Work with a partner. For each pair, one of you works out d, r and r′ while the other sketches the two circles roughly. Compare before committing to a name — if the sketch and the numbers disagree, one of you has the radii the wrong way round.
1. Choose the relationship from A to E.
A. are external to each other
B. touch externally
C. touch internally
D. one is inside the other
E. intersect at 2 points
1. Choose from A to E, as above.
(b) and (d) look almost identical, and they are not.
Both have a big circle and a small one with a short distance between the centres. In (b) the difference of the radii comes out exactly equal to d, so they touch. In (d) the difference is bigger than d, so the small circle floats free inside. One is the boundary case of the other, and only exact surds tell them apart — this is precisely where rounding would lose you the mark.
Find the equation of the circle with centre (−1, −2) that touches the circle x² + y² − 4x − 4y + 4 = 0 externally.
Not sorting the radii first.
r has to be the bigger one. Otherwise r − r′ is negative and the two internal cases collapse.
Mixing up the sum and the difference.
External touching uses r + r′. Internal touching uses r − r′. Swapping them changes the answer completely.
Rounding the surds.
3√5 − √5 = 2√5 is an exact equality. Decimals only ever suggest it, and the whole question turns on whether it is exact.
Forgetting to complete the square.
A circle in general form has no visible centre or radius. Convert it before comparing anything at all.
Reading = 9 as radius 9.
The right-hand side is r². (x − 3)² + y² = 9 has radius 3. This one costs marks in every lesson of the concept.
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