Problem
GEO-B3-M03-P004 Pole of a Given Line
B. New Original Problem. A circle has center \(O\) and radius \(R\). A line \(l\) does not pass through \(O\). Let \(H\) be the foot of the perpendicular from \(O\) to \(l\), and let \(P\) be chosen on the line \(OH\) so that \(OP\cdot OH=R^2\). Prove that \(l\) is the polar of \(P\).
C. Hint 1. The polar of \(P\) must be perpendicular to \(OP\).
D. Hint 2. It must meet \(OP\) at the point giving product \(R^2\).
E. Full Solution.
By construction \(l\perp OH\), and since \(P\) lies on \(OH\), we have \(l\perp OP\). Also, \(H=l\cap OP\) and \(OP\cdot OH=R^2\).
These two properties uniquely define the polar of \(P\) with respect to \(\omega(O,R)\): it is perpendicular to \(OP\) and meets \(OP\) at \(H\). Hence \(l\) is the polar of \(P\).
It is useful to emphasise the duality: a line has a pole once the circle is fixed.