Find the length of the perpendicular from the points \((h,k)\) on the straight line \(x\cos\alpha+y\sin\alpha=p\). Shew that \(x^2(k^2-p^2)+y^2(h^2-p^2) = 2hkxy\) is the equation of two straight lines through the origin such that the lengths of the perpendiculars on them from \((h,k)\) are each equal to \(p\).
Shew that, by a proper choice of axes, the equations of any two circles may be written in the form \(x^2+y^2+2ax+b=0\) and \(x^2+y^2+2a'x+b=0\). If the polars of a point \(P\) with respect to these two circles meet in \(Q\), shew that the axis of \(y\) bisects \(PQ\).
Find the equations of the tangent and normal to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) at the point whose eccentric angle is \(\phi\). Prove that the envelope of the line joining the points of contact of two perpendicular tangents to an ellipse is another ellipse.
Find the coordinates of the centre of a conic whose equation in trilinear coordinates is \(l\beta\gamma + m\gamma\alpha + n\alpha\beta = 0\). If this conic passes through the centre of the circle inscribed in the triangle of reference, prove that its centre lies on the conic \[ a\alpha^2+b\beta^2+c\gamma^2 - (b+c)\beta\gamma - (c+a)\gamma\alpha - (a+b)\alpha\beta = 0. \]
Find the first and second differential coefficients of \(e^{ax}\cos bx\), and deduce that the \(n\)th differential coefficient is \[ (a^2+b^2)^{n/2}e^{ax}\cos\left(bx+n\tan^{-1}\frac{b}{a}\right). \]
Explain the meanings of the partial differential coefficients \(\frac{\partial r}{\partial x}\) and \(\frac{\partial x}{\partial r}\), where \(x,y\) are the rectangular coordinates of a point and \(r, \theta\) are its polar coordinates. Prove that \(\frac{\partial^2 r}{\partial x^2} + \frac{\partial^2 r}{\partial y^2} = \frac{1}{r}\left[\left(\frac{\partial r}{\partial x}\right)^2 + \left(\frac{\partial r}{\partial y}\right)^2\right]\).
If \(p\) and \(q\) are the lengths of the perpendiculars from the origin on the tangent and normal to a curve, prove that the radius of curvature is \(p+\frac{d^2p}{d\psi^2}\), where \(\psi\) is the inclination of the tangent to the axis. Shew that in the curve \begin{align*} x &= 3a\cos t + a\cos 3t, \\ y &= 3a\sin t + a\sin 3t, \end{align*} \(p=4a\cos\left(\frac{\psi}{2}-\frac{\pi}{4}\right)\) and that the radius of curvature is \(\frac{2}{3}p\).
Homographic correspondence in Plane Geometry, with applications.
Ruled surfaces, both developable and otherwise.
Determinants.