Given two points \(A, B\), prove the existence of a system of circles with the property that the tangents from any point \(P\) on the perpendicular bisector of \(AB\) to the circles of this system are equal to \(PA\). Give a geometrical construction for the circles of the system, which (i) touch a given line, (ii) pass through a given point \(P\). Prove that, if the circle of the system through \(P\) cuts \(AP\) again in \(P'\), \(AB\) is one of the angle bisectors of \(PBP'\), and that, if the locus of \(P\) is a straight line, the locus of \(P'\) is another straight line.
If \(s=0\) is the equation of a conic, \(t=0\) the equation of one of its tangents and \(p=0\) the equation of one of its chords, interpret the equations \[ s+kpt=0, \quad s+kt^2=0, \quad s+kt=0, \] where \(k\) is a parameter. \(PP'\) is a chord of a conic \(S_o\); a conic \(S\) passes through \(P\) and has three point contact with \(S_o\) at \(P'\), and another conic \(S'\) passes through \(P'\) and has three point contact with \(S_o\) at \(P\). Prove that the other chord of intersection of \(S\) and \(S'\) is concurrent with the tangents at \(P\) and \(P'\). The tangents at \(B, C\) to a conic meet in \(A\) and the tangents at \(B', C'\) meet in \(A'\); prove that there is another conic which touches \(A'B', A'C'\) where they are cut by \(BC\), and also touches \(AB, AC\) where they are cut by \(B'C'\).
Find the conditions that the roots of \[ x^3-ax^2+bx-c=0 \] shall be (i) in G.P., (ii) in A.P., (iii) in H.P. Show that if the roots are not in A.P. then there are in general three transformations of the form \(x=y+\lambda\) such that the transformed cubic in \(y\) has its roots in G.P.
From the ordinary geometrical definitions of \(\sin x, \cos x\) and the assumption that \(\frac{d}{dx}(\sin x) = \cos x\), deduce that, if \(x\) is positive, \[ \cos x - 1 + \frac{x^2}{2!} - \dots - (-1)^m \frac{x^{2m}}{(2m)!} \] and \[ \sin x - x + \frac{x^3}{3!} - \dots - (-1)^m \frac{x^{2m+1}}{(2m+1)!} \] are positive or negative according as \(m\) is odd or even. Prove also that, if \(x^2<1\), one value of \(\tan^{-1}x\) lies between \[ x-\frac{1}{3}x^3 \quad \text{and} \quad x-\frac{1}{3}x^3+\frac{1}{5}x^5.\]
The area \(S\) and the semi-perimeter \(s\) of a triangle are fixed. Prove that for one of the sides \(a\) to be a maximum or a minimum it must be a root of the equation \[ s(x-s)x^2+4S^2=0.\] Hence show that there is one maximum and one minimum.
Two rough planes intersect at right angles in a horizontal line and make angles \(\alpha, \frac{\pi}{2}-\alpha\) (\(0 < \alpha < \frac{\pi}{4}\)) with the horizontal. Two equal rough cylinders with their axes in the same horizontal plane rest in contact with each other and each in contact with one plane. Prove that, if all the surfaces are equally rough, the coefficient of friction is not less than \[ \frac{\cos 2\alpha}{\sin\alpha+\cos\alpha+\sin 2\alpha}.\]
A circular disc of weight \(w\) and radius \(a\) can slide on a smooth vertical rod passing through a small hole at its centre. It is supported horizontally by \(n\) light vertical rods each of length \(2a\) freely hinged to it at equal intervals round its circumference, the upper ends of the rods being freely hinged to fixed supports. An elastic band of modulus \(\lambda\) and natural length \(c\) is placed round the rods. Prove that the equilibrium is stable if \[ w > \lambda n \sin\frac{\pi}{n} \left( \frac{2a}{c} n \sin\frac{\pi}{n} - 1 \right), \] and that otherwise there will be stable equilibrium when the rods are inclined to the vertical at an angle \[ \cos^{-1} \frac{c(w+\lambda n \sin\frac{\pi}{n})}{2a\lambda n^2 \sin^2\frac{\pi}{n}}.\] Assume that the band remains taut throughout and neglect friction.
A heavy elastic particle is projected from a point \(O\) at the foot of an inclined plane of inclination \(\alpha\) to the horizon. The plane through the direction of projection normal to the inclined plane meets the inclined plane in a line \(OA\) which makes an angle \(\phi\) with the line of greatest slope and the direction of projection makes an angle \(\theta\) with \(OA\). Find equations to determine the position of the particle after any number of rebounds and show that the particle will just have ceased to rebound when it again reaches the foot of the plane if \[ \tan\theta\tan\alpha = (1-e)\cos\phi,\] where \(e\) is the coefficient of restitution.
An elastic string has one end fixed at \(A\), passes through a small fixed ring at \(B\) and has a heavy particle attached at the other end. The unstretched length of the string is equal to \(\frac{1}{2}AB\). The particle is projected from any point in any manner. Assuming that it will describe a plane curve, show that the curve is in general an ellipse.
Four particles, each of mass \(m\), are connected by equal inextensible strings of length \(a\) and lie on a table at the corners of a rhombus the sides of which are formed by the strings. One of the particles receives a blow \(P\) along the diagonal outwards. Prove that the angular velocities of the strings after the blow are equal to \(P\sin\alpha/2ma\), where \(2\alpha\) (\(\alpha < \frac{\pi}{4}\)) is the angle of the rhombus at the particle which is struck.