Radical Axis - Algebraic Construction

Algebraic Construction

Referring to Figure 3, the radical axis (red) is perpendicular to the blue line segment joining the centers B and V of the two given circles, intersecting that line segment at a point K between the two circles. Therefore, it suffices to find the distance x1 or x2 from K to B or V, respectively, where x1+x2 equals D, the distance between B and V.

Consider a point J on the radical axis, and let its distances to B and V be denoted as d1 and d2, respectively. Since J must have the same power with respect to both circles, it follows that

d_{1}^{2} - r_{1}^{2} = d_{2}^{2} - r_{2}^{2}

where r1 and r2 are the radii of the two given circles. By the Pythagorean theorem, the distances d1 and d2 can be expressed in terms of x1, x2 and L, the distance from J to K

L^{2} + x_{1}^{2} - r_{1}^{2} = L^{2} + x_{2}^{2} - r_{2}^{2}

By cancelling L2 on both sides of the equation, the equation can be written

x_{1}^{2} - x_{2}^{2} = r_{1}^{2} - r_{2}^{2}

Dividing both sides by D = x1+x2 yields the equation

x_{1} - x_{2} = \frac{r_{1}^{2} - r_{2}^{2}}{D}

Adding this equation to x1+x2 = D yields a formula for x1

2x_{1} = D + \frac{r_{1}^{2} - r_{2}^{2}}{D}

Subtracting the same equation yields the corresponding formula for x2

2x_{2} = D - \frac{r_{1}^{2} - r_{2}^{2}}{D}

Read more about this topic:  Radical Axis

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