Radical Axis :
`text(Radical Axis :)`
The radical axis of two circles is the locus of a point from which the tangent segments to the two circles
are of equal length.
`text(Equation of 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 = x^2 + y^2 + 2gx +2fy+c = 0` and
`S' = x^2 + y^2 + 2g'x + 2f'y + c' = 0`
`text(Fundas :)`
`(i)` If two circles intersect, then the radical axis is the common chord of the two circles.
`(ii)` If two circles touch each other then the radical axis is the common tangent of the two circles at the common point of contact.
`(iii)` Radical axis is always perpendicular to the line joining the centres of the two circles.
`(iv)` Radical axis need not always pass through the mid point of the line joining the centres of the two circles.
`(v)` Radical axis bisects a common tangent between the two circles.
`(vi)` Pairs of circle which do not have radical axis are concentric.
`quad quad quad quad quad quad quad quad quad quad S_1 = x^2 + y^2 + 2gx +2fy+c_1 = 0`
`quad quad quad quad quad quad quad quad quad quad S _2= x^2 + y^2 + 2gx +2fy+c_2 = 0`
`(vi)`If one circle is contained in another circle then radical axis passes outside to both the circles.
`text(The Radical Axes of Three Circles :)`
Let the equations of the three circles `S_1, S_2` and `S_3` be
`S_1 equiv x^2 + y^2 + 2g_1 x + 2f_1 y + c_1 = 0`, ...... (1)
`S_2 equiv x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0` ...... (2)
and `S_3 equiv x^2 + y^2 + 2g_3x + 2f_3y + c_3 = 0` ...... (3)
Now, by the previous section, the radical axis of `S_1` and `S_2` is obtained by substracting the equations of
these circles; hence it is
`S_1 - S_2 =0` ...... (4)
Similarly, the radical axis of `S_2` and `S_3` is
`S_2-S_3=0` ...... (5)
The lines `(4)` and `(5)` meet at a point whose coordinate say, `(X, Y)` satisfy
`S_1 - S_2 = 0` and `S_2 - S_3 = 0`
hence the coordinates `(X, Y)` satisfy
`(S_1-S_2) + (S_2-S_3) =0;`
that is, `(X, Y)` satisfy
`S_1-S_3=0`..............(6)
But `(6)` is the radical axis of the circles `S_1` and `S_3` and hence three radical axes are concurrent. The point of concurrency of the three radical axes is called the radical centre.
The radical axis of two circles is the locus of a point from which the tangent segments to the two circles
are of equal length.
`text(Equation of 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 = x^2 + y^2 + 2gx +2fy+c = 0` and
`S' = x^2 + y^2 + 2g'x + 2f'y + c' = 0`
`text(Fundas :)`
`(i)` If two circles intersect, then the radical axis is the common chord of the two circles.
`(ii)` If two circles touch each other then the radical axis is the common tangent of the two circles at the common point of contact.
`(iii)` Radical axis is always perpendicular to the line joining the centres of the two circles.
`(iv)` Radical axis need not always pass through the mid point of the line joining the centres of the two circles.
`(v)` Radical axis bisects a common tangent between the two circles.
`(vi)` Pairs of circle which do not have radical axis are concentric.
`quad quad quad quad quad quad quad quad quad quad S_1 = x^2 + y^2 + 2gx +2fy+c_1 = 0`
`quad quad quad quad quad quad quad quad quad quad S _2= x^2 + y^2 + 2gx +2fy+c_2 = 0`
`(vi)`If one circle is contained in another circle then radical axis passes outside to both the circles.
`text(The Radical Axes of Three Circles :)`
Let the equations of the three circles `S_1, S_2` and `S_3` be
`S_1 equiv x^2 + y^2 + 2g_1 x + 2f_1 y + c_1 = 0`, ...... (1)
`S_2 equiv x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0` ...... (2)
and `S_3 equiv x^2 + y^2 + 2g_3x + 2f_3y + c_3 = 0` ...... (3)
Now, by the previous section, the radical axis of `S_1` and `S_2` is obtained by substracting the equations of
these circles; hence it is
`S_1 - S_2 =0` ...... (4)
Similarly, the radical axis of `S_2` and `S_3` is
`S_2-S_3=0` ...... (5)
The lines `(4)` and `(5)` meet at a point whose coordinate say, `(X, Y)` satisfy
`S_1 - S_2 = 0` and `S_2 - S_3 = 0`
hence the coordinates `(X, Y)` satisfy
`(S_1-S_2) + (S_2-S_3) =0;`
that is, `(X, Y)` satisfy
`S_1-S_3=0`..............(6)
But `(6)` is the radical axis of the circles `S_1` and `S_3` and hence three radical axes are concurrent. The point of concurrency of the three radical axes is called the radical centre.
