Two Circles

Decide whether two circles cross, touch or do not meet, find where they cross, and find the equation of their common chord.

In the exam for IGCSE Additional Maths 0606 · 4 real past-paper questions below

اقرأ هذا الدرس مشروحاً بالعربي ←

How two circles can sit

Two Wi-Fi routers each cover a circle. Do the circles overlap, just touch, or leave a gap? You can answer without drawing anything.

Everything depends on one length: the distance between the two centres. Call it d. Compare it with the two radii, $r_{1}$ and $r_{2}$ .

Formula

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

$(x_{1}, y_{1})$ — the centre of the first circle.
$(x_{2}, y_{2})$ — the centre of the second circle.
$d$ — the distance between the centres. This is Pythagoras, from the line segment lesson. Learn it: it is not on the list of formulas.

$d > r_{1} + r_{2}$ — apart: they do not meet.
$d = r_{1} + r_{2}$ — they touch at one point, outside each other.
$∣ r_{1} - r_{2} ∣ < d < r_{1} + r_{2}$ — they cross at two points.
$d = ∣ r_{1} - r_{2} ∣$ — they touch at one point, one inside the other.
$d < ∣ r_{1} - r_{2} ∣$ — one is inside the other: they do not meet.

Memory trick

Picture two coins. Slide them apart until the gap is exactly the sum of the radii: they just touch. Any further and they separate.

Key idea

John Venn drew overlapping circles to show logic in 1880. Today we call them Venn diagrams, and each overlap is two circles crossing at two points.

Key idea

Find both centres, both radii and d. Then compare d with the sum and the difference of the radii.

Do they cross, touch or miss?

Exam questions say “Show that the circles touch” or “Show that they do not intersect”. Find d, find the radii, compare, and write a conclusion.

Worked example

1
Show that $(x - 1)^{2} + (y - 2)^{2} = 9$ and $(x - 7)^{2} + (y - 10)^{2} = 49$ touch.
2
$C_{1} (1, 2), r_{1} = 3; C_{2} (7, 10), r_{2} = 7$ — Read the centres and radii.
3
$d = 6^{2} + 8^{2} = 100 = 10$ — Distance between the centres.
4
$r_{1} + r_{2} = 3 + 7 = 10$ — Sum of the radii.
5
$d = r_{1} + r_{2}$ — So the circles touch at one point, outside each other.

Worked example

1
Decide how $x^{2} + y^{2} = 1$ and $x^{2} + y^{2} - 8 x - 6 y - 11 = 0$ sit.
2
$C_{1} (0, 0), r_{1} = 1$ — No brackets, so the centre is the origin.
3
$C_{2} (4, 3), r_{2} = 16 + 9 + 11 = 6$ — Halve and flip for the centre. Then g² + f² − c.
4
$d = 4^{2} + 3^{2} = 5$ — Distance between the centres.
5
$r_{2} - r_{1} = 6 - 1 = 5 = d$ — d equals the difference of the radii.
6
$They touch, the small circle inside the big one$ — The conclusion.

Rule

No calculator on 0606 or 4037 Paper 1? Compare squares instead of square roots. If $d^{2} > (r_{1} + r_{2})^{2}$ , then $d > r_{1} + r_{2}$ .

Common mistake

Saying “they meet” just because $d < r_{1} + r_{2}$ . Also check the difference: a small circle can sit inside a big one and never meet it.

Key idea

Two numbers to compare with d: the sum of the radii and the difference. Then state what that means.

Where two circles cross

To find where two circles cross, you need points that fit both equations. Here is the trick: write both in expanded form, then subtract one from the other.

Both have $x^{2} + y^{2}$ with 1 in front, so those terms cancel. What is left is a straight line. Both crossing points lie on it. It is the line through them: the common chord.

Steps

1
1. Expand both equations so each starts $x^{2} + y^{2}$ .
2. Subtract one from the other. The squares cancel and leave a line: the common chord.
3. Make x or y the subject of the chord.
4. Substitute into either circle and solve the quadratic.
5. Use the chord to find the other coordinate of each point.

Worked example

1
Find where $x^{2} + y^{2} - 2 x - 4 y - 5 = 0$ and $x^{2} + y^{2} - 14 x - 10 y + 49 = 0$ cross.
2
$12 x + 6 y - 54 = 0$ — First minus second: −2x − (−14x) = 12x, −4y − (−10y) = 6y, −5 − 49 = −54.
3
$2 x + y = 9$ — Divide by 6. This is the common chord.
4
$y = 9 - 2 x$ — Make y the subject.
5
$x^{2} + (9 - 2 x)^{2} - 2 x - 4 (9 - 2 x) - 5 = 0$ — Substitute into the first circle.
6
$5 x^{2} - 30 x + 40 = 0$ — Expand and collect.
7
$x^{2} - 6 x + 8 = 0 \Rightarrow x = 2 or x = 4$ — Divide by 5 and factorise.
8
$(2, 5) and (4, 1)$ — Use y = 9 − 2x.

The mark scheme rewards each stage: subtracting, the chord equation, substituting, the quadratic, then the points. Show every line.

Common mistake

Subtracting while the equations are still in bracket form. The squares only cancel when both are expanded.

Common mistake

Sign slips when you subtract. Take away every term, including the numbers on their own. Check that each point fits both circles.

Key idea

Subtract the two circles and the squares disappear. The line left over is the common chord, and both crossing points are on it.

Using the common chord

The common chord has one more useful fact. The line joining the two centres cuts the common chord in half, at a right angle.

Rule

The line of centres is the perpendicular bisector of the common chord.

Check it with the last example. The centres are $(1, 2)$ and $(7, 5)$ , so the line of centres has gradient $\frac{1}{2}$ . The chord $2 x + y = 9$ has gradient $- 2$ . The product is $- 1$ , so they are perpendicular.

Some questions give you the chord and one circle, then ask about the other circle. Find the crossing points from the circle you know, then use them for the circle you do not know.

Worked example

1
Circle $C_{1}$ is $(x - 1)^{2} + (y - 2)^{2} = 10$ . Circle $C_{2}$ has centre $(7, 5)$ . Their common chord is $2 x + y = 9$ . Find the radius of $C_{2}$ .
2
$(x - 1)^{2} + (7 - 2 x)^{2} = 10$ — Substitute y = 9 − 2x into C₁. Then 9 − 2x − 2 = 7 − 2x.
3
$5 x^{2} - 30 x + 40 = 0 \Rightarrow x = 2 or 4$ — Expand, collect and solve.
4
$(2, 5) and (4, 1)$ — The crossing points. They are on C₂ as well.
5
$r_{2}^{2} = (7 - 2)^{2} + (5 - 5)^{2} = 25$ — Distance squared from the centre of C₂ to (2, 5).
6
$r_{2} = 5$ — Check with (4, 1): 3² + 4² = 25 too.

The length of the common chord is the distance between the two crossing points.

Worked example

1
Find the length of the common chord in this example.
2
$(4 - 2)^{2} + (1 - 5)^{2} = 20$ — Distance from (2, 5) to (4, 1).
3
$= 25$ — Simplify the surd.

Common mistake

Thinking the common chord is parallel to the line of centres. It crosses it at a right angle.

Key idea

The crossing points belong to both circles. Find them from one circle, and they give you facts about the other.

Which method? Read the question

Read what the question asks before you start.

You see	You do
“Show that the circles touch” or “do not meet”	Find d and both radii, compare with the sum and difference
“Find the points of intersection”	Expand, subtract for the chord, substitute into a circle
“Find the equation of the common chord”	Expand both, subtract
The chord and one circle are given	Chord into the known circle, then use the points
“Length of the common chord”	Distance between the two crossing points
No calculator	Compare $d^{2}$ with $(r_{1} + r_{2})^{2}$ and $(r_{1} - r_{2})^{2}$

Common mistake

Forgetting the conclusion. After comparing numbers, say in words whether the circles cross, touch or do not meet.

Key idea

For two circles, think distance between centres for how they sit, and subtract the equations for where they cross.

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Real past-paper questions on two circles

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