Problem
GEO-B2-M03-P004 Where the Radical Axis Meets the Line of Centres
#4
★★☆☆☆ Level 2 of 5
Circles have centres \(O_1,O_2\), with \(O_1O_2=13\), and radii \(5\) and \(8\). The radical axis meets \(O_1O_2\) at point \(H\). Find \(O_1H\).
Let \(O_1H=x\), then \(O_2H=13-x\). Write equality of powers.
Let \(O_1H=x\). Then \(O_2H=13-x\). Equality of powers gives \(x^2-5^2=(13-x)^2-8^2\). We get \(x^2-25=x^2-26x+169-64\), hence \(-25=105-26x\), so \(x=5\). Therefore \(O_1H=5\).
A numerical problem helps the student see the algebraic nature of the radical axis.