Radical Axis of two circles

The radical axis of two circles is the locus of a point which moves in such a way that the tangent segments drawn from it to the two circles are of equal length.

Equation to the Radical Axis:

In general S – S’ = 0 represents the equation of the Radical Axis to the two circles

i.e. 2x(g – g ‘) + 2y(f – f’) + c – c’ = 0

where S ≣ x2 + y2 + 2gx + 2fy + c = 0 and S’ ≣ x2 + y2 + 2g’x + 2f’y + c’ = 0

∎ If S = 0 and S’ = 0 intersect in two real and distinct points then S – S’ = 0 is the equation of the common chord of the two circles.

∎ If S’ = 0 and S = 0 touch each other (internally or externally), then S – S’ = 0 is the equation of the common tangent to the two circles at the point of contact.

Note:

∎ To find the equation of the radical axis of two circles, first make the coefficients of x2 and y2 in the equation of the two circles equal to unity.

Length of the common chord:

From figure, PQ = 2PR = 2√(AP)2-(AR)2 , where AP is the radius of the circle, S = 0 and AR is the length of the perpendicular from A to the common chord PQ.

Notes:

∎ The length of the common chord of the two circles becomes maximum when it is a diameter of the smaller one between them.

∎ If the length of the common chord is zero, then the two circles touch each other and the common chord becomes the common tangent to the two circles at the common point of contact.

Properties of Radical Axis

∎ The radical axis of two circles is perpendicular to the line joining their centres.♦

∎ Radical centre: The radical axis of three circles taken in pairs meet at a point, called the radical centre of the circles.
Coordinates of radical centre can be found by solving the equations S1 = S2 = S3 = 0.

∎ The radical centre of three circles described on the sides of a triangle as diameters is the orthocentre of the triangle.

∎ If two circles cut a third circle orthogonally, then the radical axis of the two circles pass through the centre of the third circle.
∎ The radical axis of the two circles will bisect their common tangents.

Example : Find the coordinates of the points at which the circles x2 + y2 – 4x – 2y = 4 and x2 + y2 − 12x − 8y = 12 touch each other. Find the coordinates of the point of contact and equation of the common tangent at the point of contact.

Solution: The given circles are

S1 ≣ x2 + y2 − 4x − 2y − 4 = 0 and S2 ≣ x2 + y2 − 12x − 8y − 12 = 0

Equation of the common tangent is S1 − S2 = 0 i.e. 8x + 6y + 8 = 0
4x + 3y + 4 = 0 . . . (1)

Let the point of contact be (x1, y1)

Equation of the tangent at (x1, y1) is ,

xx1 + yy1 − 2(x + x1) − (y + y1) − 4 = 0

or (x1 − 2)x + (y1 − 1)y − 2x1 − y1 − 4 = 0 . . . (2)

Comparing (1) and (2), we get

$\large \frac{x_1 -2}{4} = \frac{y_1 -1}{3} = \frac{2x_1 + y_1 + 4}{-4} = k (say)$

x1 = 4k + 2, y1 = 3k + 1 , 2x1 + y1 + 4 = − 4k

2(4k + 2) + 3k + 1 + 4 = − 4k

or 8k + 4 + 3k + 5 = − 4k

15k = −9 => k = −9/15 = −3/5

x1 = 4 x(−3/5)+2 , y1 = 3 x (−3/5)+ 1

Point of contact $\large (\frac{-2}{5} , \frac{-4}{5}) $

Example : (a) If two circles cut a third circle orthogonally, prove that their common chord will pass through the centre of the third circle.

(b) A and B are two fixed points and P moves such that PA = n PB  ; where n ≠ 1. Show that locus of P is a circle and for different values of n all the circles have a common radical axis.

Solution: (a) Let us take the equation of the two circles as
x2 + y2 + 2g1x + 2f1y + c1= 0 ….(1)

x2 + y2 + 2g2x + 2f2y + c2 = 0. ….(2)

Let the third circle be

x2+ y2 + 2gx + 2fy + c = 0. …(3)

Since (1) and (2) cut the third circle orthogonally

2g1g + 2f1f = c1 + c …(4)

and 2g2g + 2f2f = c2 + c ….(5)

(5) − (4)
2 (g2 − g1)g + 2 (f2 − f1f = c2 − c1 …(6)

Also radical axis of (1) and (2) is

2 (g1 − g2)x + (f1 − f2)y = c2 − c1 ….(7)

Put (− g , − f) in (7), we get

2 (g2 − g1)g + 2 (f2 − f1)f = c2 − c1

Hence common chord of (1) and (2) pass through the centre of the third circle.

(b) Let A ≣ (a, 0), B ≣ (−a, 0) and P(h , k)

so PA = n PB

( h − a)2 + k2 = n2 [(h + a)2 + k2]

( 1 − n2)h2 + (1 − n2)k2 − 2ah (1 + n2) + (1 − n2)a2 = 0

$\large h^2 + k^2 – 2 a h (\frac{1+n^2}{1-n^2}) + a^2 = 0 $

Hence locus of P is

$\large x^2 + y^2 – 2 a x (\frac{1+n^2}{1-n^2}) + a^2 = 0 $ , which is a circle for different values of n.

Let n1 and n2 are two different values of n so their radical axis is x = 0 i.e. y-axis. Hence for different values of n the circles have a common radical axis.

Also Read :

Equations of Circle in various forms
Parametric Equation of a Circle
Position of a Point w.r.t a Circle
Equation of Tangent & Normal on the Circle
Chord of Contact
Equation of Family of Circles
External & Internal Contacts of Circles

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