Problem
GEO-B2-M04-P008 External Centre of Two Circles
#8
★★★☆☆ Level 3 of 5
Two circles have centres \(O_1\) and \(O_2\), radii \(4\) and \(10\), and \(O_1O_2=18\). Find the distance from \(O_1\) to the external centre of homothety of the two circles.
The external centre divides the line of centres externally in the ratio of the radii.
Let \(H\) be the external centre of homothety, with the smaller circle closer to \(H\). Then \(\frac{HO_1}{HO_2}=\frac{4}{10}=\frac{2}{5}\). Since the centre is external, points \(H,O_1,O_2\) are collinear and \(HO_2=HO_1+18\). Let \(HO_1=x\). Then \(\frac{x}{x+18}=\frac{2}{5}\). Hence \(5x=2x+36\), so \(x=12\). Therefore \(HO_1=12\).
The problem trains the distinction between internal and external division.