Radical Axis and it’s Properties

Radical Axis and it’s Properties

Radical Axis: The radical axis of two circles is the locus of the point which moves such that the lengths of the tangents drawn from it to the two circles are equal.


S ≡ x² + y² + 2gx+ 2fy +c = 0 … (1)

and S ≡ x² + y² + 2g₁x+ 2f₁y + c₁ = 0 … (2)

Let P (x₁, y₁) be a point such that |PA| = |PB|

i.e., \(\sqrt{{{x}_{1}}^{2}+{{y}_{1}}^{2}+2g{{x}_{1}}+2f{{y}_{1}}+c}=\sqrt{{{x}_{1}}^{2}+2{{g}_{1}}{{x}_{1}}+2{{f}_{1}}{{y}_{1}}+{{y}_{1}}^{2}}\).

Radical Axis

On squaring in both sides

 x₁² + y₁² + 2gx₁ + 2fy₁ + c = x₁² + y₁² + 2g₁x₁ + 2f₁y₁ + c₁

2 (g – g₁) x₁ + 2 (f – f₁) y₁ + c – c₁ = 0

Which is the required equation of the radical axis of the given circles. Clearly, this is a straight line.

Properties of the Radical Axis:

1. The Radical Axis is Perpendicular to the Line joining the Centers of the given Circles:

Slope of \({{C}_{1}}{{C}_{2}}=\frac{-{{f}_{1}}+f}{-{{g}_{1}}+g}=\frac{f-{{f}_{1}}}{g-{{g}_{1}}}\) = m₁ (say).

Slope of radical axis = \(-\frac{(g-{{g}_{1}})}{f-{{f}_{1}}}\) = m₂ (say).

∴ m₁m₂ = -1.

Hence, C₁C₂ and radical axis are perpendicular to each other.

2. Radical Axis Bisects the Common Tangent of Two Circles: Let QR be the common tangent. If it meets the radical axis P, then PQ and PR, are two tangents to the circles.

Hence, PQ = PR, since the lengths of tangents are equal from any point on the radical axis. Hence, the radical axis bisects the common tangent QR.