Problem
GEO-B3-M03-P002 Polar in Coordinates
B. New Original Problem. For the unit circle \(x^2+y^2=1\), a point \(P(p,q)\), not the origin, is given. Prove that its polar has equation \(px+qy=1\).
C. Hint 1. Check that this line is perpendicular to \(OP\).
D. Hint 2. Find its intersection with \(OP\) and check the product of distances.
E. Full Solution.
The line \(px+qy=1\) has normal vector \((p,q)\), so it is perpendicular to \(OP\). Let \(H\) be its intersection with \(OP\). Then \(H=t(p,q)\). Substituting into the line equation gives \(t(p^2+q^2)=1\), hence \(t=\frac{1}{p^2+q^2}\).
We have \(OP=\sqrt{p^2+q^2}\), and \(OH=t\sqrt{p^2+q^2}=\frac{1}{\sqrt{p^2+q^2}}\). Therefore \(OP\cdot OH=1\), which defines the polar with respect to the unit circle.
The coordinate formula is useful for proving La Hire quickly and checking complex configurations.