Systems of Circles: Quick Revision

A brief note on Systems of Circles. Read it before a test. Try each question at the end before you open its answer.

Two circles

• Cases: compare $d$ with $r_1 + r_2$ and $|r_1 - r_2|$. The number of common tangents is $4$ ($d$ larger than the sum), $3$ (equal to the sum), $2$ (between), $1$ (equal to the difference), $0$ (smaller).
• Direct tangent: $\sqrt{d^2 - (r_1 - r_2)^2}$.
• Transverse tangent: $\sqrt{d^2 - (r_1 + r_2)^2}$.
• Centres of similitude: they divide $C_1C_2$ externally and internally in $r_1 : r_2$. The two pairs of tangents meet there.
• Common chord: $S_1 - S_2 = 0$. For touching circles it is the common tangent at the contact point.

Radical axis and orthogonal circles

• Radical axis: $S_1 - S_2 = 0$. Equal tangent lengths from every point of it. It is perpendicular to the line of centres.
• Radical centre: the point where the three radical axes meet. There is none if the centres are collinear.
• Orthogonal circles: $d^2 = r_1^2 + r_2^2$, or $2g_1g_2 + 2f_1f_2 = c_1 + c_2$.

Families and loci

• Circle and line: $S + \lambda L = 0$.
• Two circles: $S_1 + \lambda S_2 = 0$, and $\lambda = -1$ gives the radical axis.
• Touching a line at a point: $(x - x_1)^2 + (y - y_1)^2 + \lambda L = 0$.
• Apollonius circle: $PA : PB = m : n$ with $m \ne n$ is a circle.
• Constant difference or sum of distances: a hyperbola or an ellipse. A point as far from a point as from a line gives a parabola.

A worked example

Two circles have radii $4$ and $1$, and their centres are $13$ apart. Find the lengths of the common tangents.

1. The case. $4 + 1 = 5 < 13$, so there are $4$ common tangents.
2. Direct. $\sqrt{169 - 9} = \sqrt{160} \approx 12.65$.
3. Transverse. $\sqrt{169 - 25} = 12$.

Practice with new numbers

A program to try

The program finds the radical axis of two circles and checks that a point of it has equal powers.

Mistakes to avoid

• Mixing up the two tangent formulas. Direct uses the difference, transverse uses the sum.
• Adding the circles. The radical axis is $S_1 - S_2$.
• Wrong constant for $x^2 + y^2 = 9$. It is $c = -9$.
• No radical centre for collinear centres.

Check yourself

1. How many common tangents do circles with radii $4$ and $3$ and $d = 7$ have?

2. Find the radical axis of $x^2 + y^2 - 2x = 0$ and $x^2 + y^2 - 4y = 0$.

3. State the condition for two circles to be orthogonal.

4. Write the family of circles through the meeting points of $x^2 + y^2 = 9$ and $x = 1$.

5. What is the locus of $P$ with $PA = 3PB$?

Sources

• OpenStax, Precalculus 2e, Chapter 12 (Analytic Geometry).
• NCERT, Mathematics for Class XI, Chapter 10 (Conic Sections). The topics of this note beyond one circle belong to the JEE Main syllabus.