Derive the polar equation of a plane curve whose tangent is inclined at a constant angle \(\alpha\) to the radius vector from \(O\). Prove that the length \(d\) of a chord subtending an angle \(\beta\) at \(O\) is given by \[ d = r(1-2e^{\beta\cot\alpha}\cos\beta+e^{2\beta\cot\alpha})^{\frac{1}{2}}, \] where \(r\) is the radius vector to the end of the chord nearer to \(O\). Prove also that the line joining a point of the curve to its centre of curvature subtends a right angle at \(O\).