A radical axis of two circles is the locus of a point that moves in such a way that the tangent lines drawn from it to the two circles are of the same length. The radical axis of 2 circles is a line perpendicular to the line joining the centres. Consider two circles, S1 and S2, with centres c1 and c2. Let P be a point such that PA = PB. Then, the locus of point P is the radical axis.
Equation of the Radical Axis
The equation of the radical axis of two circles, S and S’, is given by
S – S’ = 0
i.e., 2x(g – g’) + 2y(f – f’) + c – c’ = 0.
Let S: x2 + y2 + 2gx + 2fy + c = 0 and S’: x2 + y2 + 2g’x + 2f’y + c’ = 0.
The equation of the radical axis is S – S’ = 0
x2 + y2 + 2gx + 2fy + c – (x2 + y2 + 2g’x + 2f’y + c’) = 0
=> 2x(g – g’) + 2y(f – f’) + c – c’ = 0.
a. If the two circles S and S’ intersect at real and distinct points, then the equation of the common chord is given by S – S’ = 0
b. If S’ = 0 and S = 0 touch each other, then the equation of the common tangent to the two circles at the point of contact is given by S – S’ = 0.
Properties
1. The radical axis and the common chord are identical.
2. The straight line joining the centres of two circles is perpendicular to the radical axis.
3. The radical axis bisects the common tangents.
4. The radical axis of three circles taken in pairs is concurrent.
5. The radical axis of the two circles will pass through the centre of the third circle if two circles cut a third circle orthogonally.
6. The radical axis of three circles taken two at a time are concurrent, and the point of concurrency is known as the radical centre.
7. The pair of circles which do not have a radical axis are concurrent.
8. A system of circles is called a coaxial system when every pair of circles have the same radical axis.
Also, Read:
Circles and System of Circles Previous Year Questions with Solutions
Solved Examples
Example 1:
The equation of the radical axis of the two circles 7x2 + 7y2 – 7x + 14y + 18 = 0 and 4x2 + 4y2 – 7x + 8y + 20 = 0 is
a. 3x2 + 3y2 – 6y – 2 = 0
b. 21x – 68 = 0
c. x – 2y – 5 = 0
d. x + 2y + 5 = 0
Solution:
Let S1: 7x2 + 7y2 – 7x + 14y + 18 = 0
Divide the equation by 7.
=> x2 + y2 – x + 2y + 18/7 = 0
S2: 4x2 + 4y2 – 7x + 8y + 20 = 0
Divide the equation by 4.
x2 + y2 – (7/4)x + 2y + 5 = 0
The equation of the radical axis is S1 – S2 = 0
=> x2 + y2 – x + 2y + 18/7 – (x2 + y2 – (7/4)x + 2y + 5) = 0
=> (3/4)x – 17/7 = 0
=> 21x – 68 = 0
Hence, option b is the answer.
Example 2:
Show that the circle x2 + y2 – 6x – 4y + 9 = 0 bisects the circumference of the circle x2 + y2 – 8x – 6y + 23 = 0.
Solution:
Let S1: x2 + y2 – 6x – 4y + 9 = 0
S2: x2 + y2 – 8x – 6y + 23 = 0
The equation of the radical axis is S1 – S2 = 0
=> 2x + 2y – 14 = 0
=> x + y – 7 = 0…(i)
Centre of circle S2 = (4, 3)
Centre (4, 3) satisfies (i).
So, we can say that line (i) passes through the centre (4, 3).
Line (i) is the diameter of circle S2. Hence, S1 bisects the circumference of the circle S2.
Hence proved.
Radical Axis – Video Lesson
Frequently Asked Questions on Radical Axis
What do you mean by the radical axis of two circles?
The locus of a point that moves in such a way that the tangent lines drawn from it to the two circles are of the same length is called the radical axis of two circles.
Is the radical axis perpendicular to the common tangent?
The radical axis is perpendicular to the straight line joining centres of two circles.
Give the equation for the radical axis.
The equation of the radical axis of two circles S and S’ is given by S – S’ = 0
i.e., 2x(g – g’) + 2y(f – f’) + c – c’ = 0.
Does a radical axis exist for two concentric circles?
Concentric circles have no real radical axis.
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