Establish that the radius of curvature of a plane curve whose pedal equation is \(r=r(p)\) is \(r dr/dp\). Show that the square of the length of the tangent from the pole to the circle of curvature at a general point is given by \(\frac{d}{dq}(r^2q)\), where \(q=1/p\). Hence show that if a curve is such that all its circles of curvature pass through a certain fixed point, then it must itself be a circle.