Two pairs of points \(A, B\) and \(A', B'\) lie on an axis \(Ox\), and their abscissae are given by the equations \(ax^2+2bx+c=0\) and \(a'x^2+2b'x+c'=0\) respectively. Find an equation with rational coefficients which has \(AA' \cdot BB'\) for one of its roots. Give the geometrical interpretation of the relations obtained by equating the various coefficients in the equation to zero.
Three spherical balls, two of which have a radius of 1 inch and the third a radius of 2 inches, rest on a table, the points of contact being corners of an equilateral triangle of side 6 inches. A fourth ball rests on the table and touches each of the other three; prove that its radius is slightly greater than 2.4 inches.
In a quadrilateral \(ABCD\) the sides are \(AB=a, BC=b, CD=c, DA=d\); and the angle \(DAB=\theta, ABC=\phi\). Prove that \[ 2bd \cos(\theta+\phi) - 2ad \cos\theta - 2ab \cos\phi + a^2+b^2+d^2-c^2 = 0. \] Shew also that, if the quadrilateral is slightly deformed so that its sides remain of constant length, \[ \frac{\delta A}{\Delta BCD} = -\frac{\delta B}{\Delta CDA} = \frac{\delta C}{\Delta ABD} = -\frac{\delta D}{\Delta ABC}. \]
Discuss the general form of the curve \(y=x-a \log(x/b)\), where \(a\) and \(b\) are positive, and give a rough sketch of the curve. Find the asymptote. Prove that at any point \(P\) the chord of curvature, parallel to the asymptote, is proportional to the square of the length \(PT\) of the tangent intercepted between \(P\) and the asymptote. Give geometrical constructions for the point \(T\) and for the centre of curvature at \(P\).
Shew that, if by inversion in a plane three given points are inverted into three points forming the vertices of an equilateral triangle, the origin of inversion has two possible positions. Extend the result to the case of possible origins not in the plane of the original triangle and shew that they lie on a circle of which the line, joining the two origins in the plane, is the diameter.
The asymptotes of each of two rectangular hyperbolas are parallel to the axes of the other, and each hyperbola passes through the centre of the other. Prove that the normals to each hyperbola at its points of intersection with the other are concurrent, and that the centres of the hyperbolas are the points of trisection of the line joining the two points of concurrence.
Prove that the two conics, which pass through the four corners of a given square and touch a given line, are real unless the given line separates one corner of the square from the other three corners. State the analogous theorem in regard to the two conics touching the sides of a given square and passing through a given point.
Two equal uniform smooth cylinders, of radius \(a\), rest in a horizontal cylindrical groove of radius \(b\). A third cylinder, equal in all respects to the first two, is placed upon them. Shew that the two lower cylinders will just separate if \((b-a)^2 = 28a^2\).
A tetrahedron \(ABCD\) is formed of light rods smoothly jointed at their extremities and \(X, Y\), the middle points of \(AB, CD\), are joined by a string in which there is a tension \(T\). Prove that the tension in \(AB\) is \(\frac{1}{4}T \cdot \frac{AB}{XY}\) and write down the stresses in the other rods, stating in each case whether the stress is a tension or a thrust.
Two imperfectly elastic particles of equal mass, whose coefficient of restitution is \(e\), are suspended from the same point by light strings of equal length. One particle is drawn aside a small distance \(x_0\), and then released. Shew that, between the \(n\)th and \((n+1)\)th impacts, the particle originally drawn aside swings through a distance \(\frac{1}{2}\{1+(-e)^n\}x_0\) on one side of the vertical through the point of suspension.