Radical axis and orthogonal circles

What you will be able to do

• find the radical axis of two circles and use its properties;
• find the radical centre of three circles;
• test whether two circles are orthogonal, and find an unknown constant from the test.

The fence between two ponds

Two round ponds in a park overlap. A path is to run along the line through the two points where their edges meet. Now suppose the ponds do not overlap. Is there still a natural line between them? Yes: the line of the points from which a rope pulled tight to each pond has the same length. This line is the radical axis.

The radical axis

Take two circles $S_1 = x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0$ and $S_2 = x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0$. From the last lesson, the number $S_1(P)$ is the power of a point $P$ about the first circle. For a point outside, it equals the square of the tangent length. The radical axis is the set of points $P$ where the two powers are equal:

So the radical axis is the straight line $S_1 - S_2 = 0$. It is perpendicular to the line of centres, because $(g_1 - g_2, f_1 - f_2)$ is along the line of centres and it is the normal of this line.

What does it look like in each case?

• Crossing circles: the radical axis is the common chord.
• Touching circles: it is the common tangent at the point of contact.
• Apart circles: it lies between them. From any point on it, outside both circles, the two tangent lengths are equal.
• Concentric circles: there is no radical axis. The subtraction leaves only a constant.

A worked example

Find the radical axis of $S_1: x^2 + y^2 + 4x - 6y + 4 = 0$ and $S_2: x^2 + y^2 - 8x - 2y + 8 = 0$. Then find the point of it on the $x$-axis and the tangent length from that point.

1. Subtract. $S_1 - S_2 = 12x - 4y - 4 = 0$, which is $3x - y - 1 = 0$.
2. Check the direction. The centres are $(-2, 3)$ and $(4, 1)$. The line of centres has slope $\dfrac{1 - 3}{4 + 2} = -\dfrac{1}{3}$. The radical axis has slope $3$. The product is $-1$, so they are perpendicular.
3. The point on the $x$-axis. Put $y = 0$: $3x = 1$, so $x = \dfrac{1}{3}$.
4. The tangent length. $S_1\left(\tfrac{1}{3}, 0\right) = \dfrac{1}{9} + \dfrac{4}{3} + 4 = \dfrac{49}{9}$. So the tangent length is $\dfrac{7}{3}$. The value of $S_2$ at the same point is also $\dfrac{49}{9}$.

The radical centre

Take three circles whose centres are not on one line. Their three radical axes meet at one point, the radical centre. The reason is short. Let $P$ be the meeting point of the axes of the first pair and the second pair. Then $S_1(P) = S_2(P)$ and $S_2(P) = S_3(P)$. So $S_1(P) = S_3(P)$, and $P$ is on the third axis as well.

From the radical centre the three tangent lengths are equal. So the circle with the radical centre as its centre and this common length as its radius cuts all three circles at right angles, as the next section shows. If the three centres are on one line, the three axes are parallel (or the same line) and there is no radical centre.

Example. Take $x^2 + y^2 + 4x + 2y - 6 = 0$, $x^2 + y^2 - 12x - 2y + 30 = 0$ and $x^2 + y^2 - 4x + 10y + 2 = 0$. The first minus the second gives $16x + 4y - 36 = 0$, which is $4x + y = 9$. The first minus the third gives $8x - 8y - 8 = 0$, which is $x - y = 1$. They meet where $5x = 10$, so at $(2, 1)$. The power there is $4 + 1 + 8 + 2 - 6 = 9$ for the first circle, and also $9$ for the others. The tangent length is $3$.

Orthogonal circles

Two circles cut orthogonally when their tangents at a meeting point are perpendicular. The radius of each circle at that point is perpendicular to its own tangent. So the two radii are also perpendicular, and the centres and the meeting point form a right triangle with hypotenuse $d$:

Write $d^2 = (g_1 - g_2)^2 + (f_1 - f_2)^2$ and $r_i^2 = g_i^2 + f_i^2 - c_i$. After simplifying, the condition becomes

Example. Find $k$ so that $x^2 + y^2 + 2x + ky - 4 = 0$ cuts $x^2 + y^2 - 4x - 6y + 9 = 0$ at right angles. Here $g_1 = -2$, $f_1 = -3$, $c_1 = 9$ and $g_2 = 1$, $f_2 = \dfrac{k}{2}$, $c_2 = -4$. The condition is $2(-2)(1) + 2(-3)\dfrac{k}{2} = 9 - 4$, so $-4 - 3k = 5$ and $k = -3$. As a check, the centres are $(2, 3)$ and $(-1, 1.5)$, so $d^2 = 9 + 2.25 = 11.25$. The radii squared are $4 + 9 - 9 = 4$ and $1 + 2.25 + 4 = 7.25$. Their sum is $11.25$. It matches.

Two useful facts follow. First, the centre of a circle that is orthogonal to two given circles has equal powers about them, so it lies on their radical axis. Second, if a circle $S$ is orthogonal to the circle $S_1$, then the power of the centre of $S$ about $S_1$ is $d^2 - r_1^2 = r^2$. So it equals the square of the radius of $S$.

Practise with new numbers

Try it in Python

The program finds the radical axis of two circles and the radical centre of three. It checks that the power of the centre is the same about all three circles.

Common mistakes

• Using $S_1 + S_2$. The radical axis is the difference $S_1 - S_2$.
• Forgetting that the circles must have the same coefficient of $x^2 + y^2$. Divide first.
• Writing the circle $x^2 + y^2 = 9$ with $c = 9$. In $x^2 + y^2 - 9 = 0$ the constant is $c = -9$.
• Expecting a radical centre when the centres are collinear. There the axes are parallel.

Remember these

• The radical axis of $S_1 = 0$ and $S_2 = 0$ is $S_1 - S_2 = 0$. It is perpendicular to the line of centres.
• Every point of the radical axis has the same power, and so the same tangent length, about both circles.
• The radical centre of three circles is where their three radical axes meet.
• Orthogonal circles: $d^2 = r_1^2 + r_2^2$, or $2g_1g_2 + 2f_1f_2 = c_1 + c_2$.

Check yourself

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

1. Find the radical axis of $x^2 + y^2 - 4x - 2y - 4 = 0$ and $x^2 + y^2 - 6x + 4y + 8 = 0$.

2. For which value of $c$ does $x^2 + y^2 - 8x + c = 0$ cut $x^2 + y^2 = 9$ at right angles?

3. What is the radical axis of two circles that touch from outside?

4. Three circles have centres on one straight line. What can you say about their radical axes?

5. A point $P$ is on the radical axis of two circles, outside both. The tangent from $P$ to the first circle has length $5$. What is the tangent length to the second circle?

References

• NCERT, Mathematics for Class XI, Chapter 10 (Conic Sections). The radical axis and orthogonal circles are in the JEE Main syllabus and in coordinate geometry textbooks, not in the NCERT text.
• OpenStax, Precalculus 2e, Chapter 12 (Analytic Geometry), for the circle.