Problem
GEO-B2-M03-P011 A Second Radical Axis Computation
#11
★★★☆☆ Level 3 of 5
The distance between the centres of two circles is \(20\), and their radii are \(13\) and \(7\). The radical axis meets the line of centres at point \(H\). Find the distance from the centre of the first circle to \(H\).
Let \(O_1H=x\), then \(O_2H=20-x\).
Equality of powers at point \(H\) gives \(x^2-13^2=(20-x)^2-7^2\). Then \(x^2-169=x^2-40x+400-49\), hence \(-169=351-40x\), so \(40x=520\) and \(x=13\). Therefore \(O_1H=13\).
The numbers are chosen so that the result is geometrically visible: the point lies on the first circle.