Prove that at a point of inflexion on a curve, \(\frac{d^2y}{dx^2}=0\); and that if \(x,y\) are functions of a parameter \(t\), \(x'y''-y'x''=0\), where dashes denote differentiation with respect to \(t\).
Find the points of inflexion on the curve \(x=2a\cos t+b\cos 2t, y=2a\sin t+b\sin 2t\), where \(a,b\) are positive and show that they are real if \(b
For a curve defined by \(p=f(\psi)\), prove that the projection of the radius vector on the tangent is \(\frac{dp}{d\psi}\) and that \(\rho = p+\frac{d^2p}{d\psi^2}\). For the curve \(p=a\sin 2\psi\), prove that \(r^2=a^2-3p^2\) and that the \(p,r\) equation of the locus of centres of curvature is \(r^2=16a^2-3p^2\).