A hyperbola may be defined as the locus of a point \(P\) whose distances from two fixed points \(A, B\) have a constant difference; from this definition deduce, without assuming any other property of the hyperbola, the following sequence of results:
The points \(P(x,y)\), \(P'(x',y')\) of a plane are said to correspond when their coordinates are connected by the relations \[ x' : y' : 1 = a(b+h)x + bhy : akx + b(a+k)y : ax+by+ab; \] shew that, if \(P\) describes a line \(\lambda\), \(P'\) describes a line \(\lambda'\). Determine all the points \(P\) which coincide with their corresponding points \(P'\) and all the lines \(\lambda\) which coincide with their corresponding lines \(\lambda'\). Prove that all corresponding pairs of points \(P, P'\) are collinear with a fixed point \(A\) and that all corresponding pairs of lines \(\lambda, \lambda'\) are concurrent with a fixed line \(a\), and shew that, if \(ah+bk+2ab=0\), every pair of corresponding points \(P,P'\) are harmonically separated by \(A\) and \(a\).
If \(\alpha, \beta, \gamma, \delta\) are the roots of the equation \[ x^4 + px^2 + qx + r = 0, \] prove that the equation whose roots are \(\beta+\gamma, \gamma+\alpha, \alpha+\beta\) is \[ y^3 + 2py^2 + (p+2s^2)y - q=0, \text{ (OCR error, likely } p^2 \text{ or similar)} \] and deduce that the equation whose roots are \(\beta+\gamma, \gamma+\alpha, \alpha+\beta, \alpha+\delta, \beta+\delta, \gamma+\delta\) is \[ z^2(z^2+p)^2 - 4rz^2 - q^2=0. \] The roots of the equation \(x^4 + 4ax^3+6bx^2+4cx+d=0\) are \(\lambda_1, \lambda_2, \lambda_3, \lambda_4\). By reducing this equation to the form \(y^4+py^2+qy+r=0\), or otherwise, prove that the equation whose roots are \[ \tfrac{1}{4}(\lambda_1+\lambda_4-\lambda_2-\lambda_3)^2, \quad \tfrac{1}{4}(\lambda_2+\lambda_4-\lambda_3-\lambda_1)^2, \quad \tfrac{1}{4}(\lambda_3+\lambda_4-\lambda_1-\lambda_2)^2 \] is \[ z(z+6b-6a^2)^2 - 4(d-4ac+6a^2b-3a^4)z - (4c-12ab+8a^3)^2 = 0. \] Deduce that, if the roots of the equation \[ x^4+4ax^3+6a^2x^2+4cx+d=0 \] are all real, then they are all equal.
Prove that, if \(a_1, a_2, \dots, a_n\) are all different, the polynomial of degree \(n-1\) which takes the values \(A_1, A_2, \dots, A_n\) when \(x=a_1, a_2, \dots, a_n\) respectively is \[ \sum_{i=1}^n A_i \frac{(x-a_1)\dots(x-a_{i-1})(x-a_{i+1})\dots(x-a_n)}{(a_i-a_1)\dots(a_i-a_{i-1})(a_i-a_{i+1})\dots(a_i-a_n)}. \] Deduce the values of \[ \sum_{i=1}^n \frac{a_i^m}{(a_i-a_1)\dots(a_i-a_{i-1})(a_i-a_{i+1})\dots(a_i-a_n)} \] for \(m=1,2,\dots, n-2, n-1\).
State how to find the differential coefficient with respect to \(x\) of \[ \int_u^v f(x,t)dt, \] where \(u, v\) are functions of \(x\) and \(f(x,t)\) is a function of \(x,t\). If \[ \psi(x) = x^n \int_0^x \psi(t) dt, \] prove that \[ x \frac{d\psi(x)}{dx} = (n+x^{n+1})\psi(x), \] and hence find the form of \(\psi(x)\).
Explain what is meant by \[ \left( x \frac{\partial}{\partial x} + y \frac{\partial}{\partial y} \right)^2 z, \] where \(z\) is a function of \(x\) and \(y\), and prove that it is not in general equal to \[ x^2 \frac{\partial^2 z}{\partial x^2} + 2xy \frac{\partial^2 z}{\partial x \partial y} + y^2 \frac{\partial^2 z}{\partial y^2}. \] If \(z=x^n f(y/x)\), prove that \[ x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = nz, \] and that \[ x^2 \frac{\partial^2 z}{\partial x^2} + 2xy \frac{\partial^2 z}{\partial x \partial y} + y^2 \frac{\partial^2 z}{\partial y^2} = n(n-1)z. \] Find the value of \[ c_0 x^p \frac{\partial^p z}{\partial x^p} + c_1 x^{p-1} y \frac{\partial^p z}{\partial x^{p-1} \partial y} + \dots + c_k x^{p-k} y^k \frac{\partial^p z}{\partial x^{p-k} \partial y^k} + \dots + c_p y^p \frac{\partial^p z}{\partial y^p}, \] where \(c_0, c_1, \dots, c_k, \dots, c_p\) are the coefficients in order in the binomial expansion of \((1-\xi)^p\).
A light rectangular rigid table, which has a leg at each corner of the top, has a particle of weight \(W\) placed on it at a point whose coordinates are \((x,y)\), referred to rectangular axes; the edges of the table are given by the equations \(x \pm a=0, y \pm b=0\). The floor is slightly elastic and gives under each leg by a small amount proportional to the pressure of the leg on the floor. Shew that the forces acting in the legs are \(\frac{1}{4}W(1\pm \frac{x}{a} \pm \frac{y}{b})\), provided that \(x\) and \(y\) are such that all these four numbers are positive.
A trolley consists of a uniform rectangular platform of length \(2c\) with two pairs of wheels of radius \(a\). The wheels are mounted on axles in the plane of the platform. The axles are both parallel to the front edge of the platform and are at a distance \(d\) in front of and behind the centre of the platform. The trolley is drawn by a light handle attached to the middle point of the front edge of the platform, and the handle slopes at an angle \(\alpha\) to the horizontal. The weight of the platform is \(W\) and the weight of the wheels is neglected. The friction at the axle is such that the couple needed to turn each pair is \(\lambda\) times the pressure on it. Find
Two identical rectangular blocks each of mass \(m\) rest on a horizontal table with two faces in contact. A wedge consisting of an isosceles triangular prism of mass \(M\) and angle \(2\alpha\) has its vertex inserted between them. If the table and the faces of the wedge are smooth, shew that the wedge will descend under its own weight with acceleration \[ \frac{Mg}{M+2m\tan^2\alpha}. \] If the table and the faces of the wedge are rough, but not sufficiently so to prevent sliding, shew that the acceleration is \[ g \frac{M\cot(\alpha+\epsilon) - \mu(M+2m)}{M\cot(\alpha+\epsilon) - \mu M + 2m \tan\alpha}, \] where \(\epsilon\) is the angle of friction at the face of the wedge and \(\mu\) is the coefficient of friction between the blocks and the table. If the height of each block is \(h\) and the length of each in the direction of motion is \(2a\), find the greatest allowable value of \(h/a\) in order that the motion may take place without the blocks tipping.
A uniform rod of mass \(M\) and length \(2l\) is freely pivoted about its centre so that it can rotate in a vertical plane under gravity. A particle of mass \(m\) is attached by a string of length \(a\) to one end of the rod. It is found that it is possible for the system to oscillate about its equilibrium position so that the rod and string remain in the same plane and make equal and opposite angles with the vertical. Shew that \(Ml = 6(a-l)m\), and find the period of the oscillation.