Shew that from any point four normals can be drawn to an ellipse, and that their feet lie on a rectangular hyperbola through the point. \par Prove that the perpendicular from the point on a diameter of the ellipse meets the conjugate diameter on this hyperbola.
Find the length of the latus rectum, and the coordinates of the focus, of the parabola \[ (x\cos\alpha+y\sin\alpha)^2 = 4a(x\cos\beta+y\sin\beta+c). \] Prove that the equation of the tangents from the origin is \[ c(x\cos\alpha+y\sin\alpha)^2 + a(x\cos\beta+y\sin\beta)^2 = 0. \]
Explain what is meant by reciprocation. \par Prove that the conic \(x^2-y^2\cos\alpha-2x\sin\alpha+1=0\) is its own reciprocal with regard to the circle \(x^2+y^2=1\).
Express \(\tan n\theta\) in powers of \(\tan\theta\), distinguishing the cases according as \(n\) is odd or even. \par Prove that \[ \tan\theta+\tan\left(\theta+\frac{\pi}{2n}\right)+\tan\left(\theta+\frac{2\pi}{2n}\right)+\dots+\tan\left(\theta+\frac{2n-1}{2n}\pi\right) = -2n\cot 2n\theta, \] and find the sum of the squares of the same tangents.
Express \(1-\cos^2\theta-\cos^2\phi-\cos^2\psi+2\cos\theta\cos\phi\cos\psi\) as the product of four sines. \par Solve the equation \(\cot^{-1}\frac{x}{a}+\cot^{-1}\frac{x}{b}+\cot^{-1}\frac{x}{c}+\cot^{-1}\frac{x}{d} = \frac{\pi}{2}\).
If \(S\) be the area of a quadrilateral whose sides are \(a,b,c,d\), prove that \[ S^2 = (s-a)(s-b)(s-c)(s-d)-abcd\cos^2\alpha, \] where \(2s=a+b+c+d\) and \(2\alpha\) is the sum of two opposite angles. \par If the quadrilateral can be inscribed in a circle, prove that the radius of the circle is \[ \frac{1}{4}\left\{\frac{(ab+cd)(ac+bd)(ad+bc)}{(s-a)(s-b)(s-c)(s-d)}\right\}^{\frac{1}{2}}. \]
If \[ \cos^{-1}(\alpha+i\beta) = A+iB, \] prove that \[ \alpha^2\sec^2A-\beta^2\operatorname{cosec}^2A=1 \] and \[ \alpha^2\operatorname{sech}^2B+\beta^2\operatorname{cosech}^2B=1. \] Prove that \(\tan^{-1}\frac{\tan 2\theta+\tanh 2\phi}{\tan 2\theta-\tanh 2\phi} + \tan^{-1}\frac{\tan\theta-\tanh\phi}{\tan\theta+\tanh\phi} = \tan^{-1}(\cot\theta\coth\phi)\).
In what sense is a couple a vector? Give reasons for your answer. \par If forces completely represented by the sides of a triangle taken in order are in equilibrium with three equal forces acting at the corners of the triangle along the tangents to the circumcircle the same way round, prove that the triangle must be equilateral.
A rod of length \(2l\) with one end on a horizontal plane leans against a circular cylinder of radius \(a\) whose axis lies in the plane, \(\lambda\) and \(\lambda'\) being the angles of friction at the two contacts. If \(\theta\) be the greatest angle the rod can make with the horizontal in such a position, shew that \[ l\sin\theta\sin(\theta+\lambda-\lambda') = a\sin\lambda\cos\lambda' \] provided this value of \(\theta\) is greater than \(\tan^{-1}(a/2l)\).
Explain how the principle of virtual work may be used to determine the unknown reactions of a system in equilibrium. \par A regular octahedron formed of twelve equal rods of weight \(w\) freely jointed is suspended from one corner. Prove that the thrust in each horizontal rod is \(3w/\sqrt{2}\).