Two circles and their common tangents

What you will be able to do

• decide how two circles lie from $d$, $r_1$ and $r_2$, and count their common tangents;
• find the length of a direct and of a transverse common tangent;
• find the point of contact of two touching circles and the centres of similitude;
• find the common chord of two crossing circles and its length.

Two coins on a desk

Put two coins flat on a desk. Slide one towards the other. At first they are apart. Then they touch. Then they overlap. Then one lies on top of the other. A straight ruler that touches both coins without cutting them is a common tangent. How many such rulers can you place? The answer depends only on how far apart the centres are.

You know single circles from the previous course. Here the new idea is simple: only three numbers matter, $d$, $r_1$ and $r_2$. Let $C_1$ and $C_2$ be the centres and let

The five cases

• $d > r_1 + r_2$. The circles are apart. They have four common tangents.
• $d = r_1 + r_2$. The circles touch from outside. They have three common tangents. Two are outer tangents, and the third is the tangent at the point of contact.
• $|r_1 - r_2| < d < r_1 + r_2$. The circles cross at two points. They have two common tangents, both outer.
• $d = |r_1 - r_2|$. One circle touches the other from inside. There is one common tangent, the one at the point of contact.
• $d < |r_1 - r_2|$. One circle lies inside the other. There is no common tangent.

The reason is the triangle inequality. The three numbers $d$, $r_1$ and $r_2$ form a triangle exactly when $|r_1 - r_2| < d < r_1 + r_2$, and then the circles cross.

Direct and transverse tangents

A common tangent that leaves both circles on the same side is a direct (outer) common tangent. One that has the circles on opposite sides, so that it crosses the line of centres between them, is a transverse (inner) common tangent.

Let a tangent touch the circles at $P_1$ and $P_2$. The radii $C_1P_1$ and $C_2P_2$ are both perpendicular to the tangent. Draw the line from $C_2$ parallel to the tangent to meet $C_1P_1$. You get a right triangle with hypotenuse $d$.

• For a direct tangent the other two sides are $P_1P_2$ and $r_1 - r_2$.
• For a transverse tangent they are $P_1P_2$ and $r_1 + r_2$.

By Pythagoras,

A transverse tangent exists only when $d \ge r_1 + r_2$, and a direct tangent only when $d \ge |r_1 - r_2|$.

Centres of similitude

The two direct tangents meet on the line of centres at a point $E$ that divides $C_1C_2$ externally in the ratio $r_1 : r_2$. The two transverse tangents meet at a point $I$ that divides $C_1C_2$ internally in the same ratio. These two points are the external and internal centres of similitude. Every pair of circles has them. The similar triangles formed by a tangent and the two radii are the reason.

For touching circles the point of contact is one of them. If the circles touch from outside, the point of contact divides $C_1C_2$ internally in the ratio $r_1 : r_2$. If they touch from inside, it divides $C_1C_2$ externally in that ratio.

A worked example

The circles $x^2 + y^2 = 9$ and $x^2 + y^2 - 24x - 10y + 165 = 0$ have centres $(0, 0)$ and $(12, 5)$ and radii $3$ and $2$.

1. The distance. $d = \sqrt{144 + 25} = 13$.
2. The case. $r_1 + r_2 = 5$ and $13 > 5$, so the circles are apart. There are four common tangents.
3. The lengths. A direct tangent has length $\sqrt{169 - 1} = \sqrt{168} \approx 12.96$. A transverse tangent has length $\sqrt{169 - 25} = 12$.
4. The centres of similitude. The internal one divides $C_1C_2$ internally in $3 : 2$. Its $x$-coordinate is $\dfrac{3 \times 12 + 2 \times 0}{5} = 7.2$ and its $y$-coordinate is $\dfrac{3 \times 5 + 2 \times 0}{5} = 3$. So $I = (7.2, 3)$. The external one divides $C_1C_2$ externally in $3 : 2$. Its coordinates are $\dfrac{3 \times 12 - 2 \times 0}{3 - 2} = 36$ and $\dfrac{3 \times 5 - 2 \times 0}{3 - 2} = 15$. So $E = (36, 15)$.

The common chord

When the circles cross, the line through the two crossing points is the common chord. Subtract the two equations $S_1 = 0$ and $S_2 = 0$ (both written with $x^2 + y^2$ and coefficient $1$). The $x^2 + y^2$ terms cancel. What is left is a straight line, $S_1 - S_2 = 0$, and both crossing points satisfy it. So this line is the common chord. For touching circles the same line is the common tangent at the point of contact.

Example. The circles $x^2 + y^2 - 4x - 6y - 12 = 0$ and $x^2 + y^2 - 16x - 6y + 48 = 0$ have centres $(2, 3)$ and $(8, 3)$ and equal radii $5$. Subtracting gives $12x - 60 = 0$, which is $x = 5$. The distance of $(2, 3)$ from this line is $3$, so the common chord has length $2\sqrt{25 - 9} = 8$.

Practise with new numbers

Try it in Python

The program takes two circles and says how they lie and how many common tangents they have. It prints the lengths of the tangents that exist.

Common mistakes

• Mixing up the two lengths. The direct tangent uses $r_1 - r_2$. The transverse tangent uses $r_1 + r_2$.
• Forgetting the case of touching. When $d = r_1 + r_2$ there are three tangents, not two or four.
• Using the formula when no such tangent exists. If $d < r_1 + r_2$ there is no transverse tangent, because the number under the root is negative.
• Subtracting circles that are not in the same form. Write both with $x^2 + y^2$ and coefficient $1$ before you subtract.

Remember these

• Compare $d$ with $r_1 + r_2$ and $|r_1 - r_2|$. The number of common tangents is $4$, $3$, $2$, $1$, $0$ in that order of decreasing $d$.
• Direct tangent: $\sqrt{d^2 - (r_1 - r_2)^2}$. Transverse tangent: $\sqrt{d^2 - (r_1 + r_2)^2}$.
• The centres of similitude divide $C_1C_2$ in the ratio $r_1 : r_2$, externally and internally.
• $S_1 - S_2 = 0$ is the common chord of crossing circles and the common tangent at the contact point of touching circles.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Two circles with radii $4$ and $3$ have their centres $10$ apart. How many common tangents do they have?

2. Two circles with radii $4$ and $3$ have their centres $7$ apart. How many common tangents do they have?

3. For radii $4$ and $3$ and $d = 10$, find the length of a direct common tangent.

4. The circle $(x - 3)^2 + y^2 = 4$ and the circle $x^2 + y^2 = 25$ touch. Where do they touch?

5. For which values of $d$ do circles with radii $4$ and $3$ cross at two points?

References

• OpenStax, Precalculus 2e, Chapter 12 (Analytic Geometry), for the circle and the distance formula.
• NCERT, Mathematics for Class XI, Chapter 10 (Conic Sections). The text treats one circle at a time. Two circles belong to the JEE Main syllabus and to coordinate geometry books.