If \(y^2 = 1+x^2\) and \(t = (x-a)/(y+b)\), where \(b^2=1+a^2\), shew that \(x\) and \(y\) can be expressed in the forms \[ x = \frac{a(1+t^2)+2bt}{1-t^2}, \quad y = \frac{b(1+t^2)+2at}{1-t^2}. \] Shew that the range of values from \(t=-1\) to \(t=+1\) corresponds to the half of the conic for which \(y\) is positive (b being supposed positive) and indicate in a sketch the variation of \(t\) along the remainder of the conic. Prove that \[ \int_a^{x_1} \frac{dx}{y} = \log\left(\frac{1+t_1}{1-t_1}\right), \text{ where } t_1 = \frac{x_1-a}{y_1+b} \text{ and } y_1, y \text{ are positive.} \] Shew also that \[ \int_a^{x_1} \frac{y-b}{x-a} \frac{dx}{y} = \log\left(\frac{1+ax_1+by_1}{2b^2}\right). \]
The angles of any triangle \(ABC\) are trisected and the two trisectors nearest to the side \(BC\) meet in \(X\), those nearest to \(CA\) in \(Y\), and to \(AB\) in \(Z\). Shew that the angles of the triangle \(AYZ\) are equal to \(\frac{1}{3}A, \frac{1}{3}(\pi+C)\) and \(\frac{1}{3}(\pi+B)\); and deduce that the triangle \(XYZ\) is equilateral, each side being equal to \(8R \sin\frac{1}{3}A \sin\frac{1}{3}B \sin\frac{1}{3}C\), where \(R\) is the radius of the circumcircle of \(ABC\).
An elliptic wire is fixed with its major axis vertical and the ends of a uniform rod of length \(2l (<2a)\) are constrained to move on the wire. Shew that if \(\theta\) is the inclination of the rod to the horizontal, the height of the centre of the rod above the centre of the ellipse is \[ a^2 \cos\theta \sqrt{\frac{1}{a^2 \cos^2\theta + b^2 \sin^2\theta} - \frac{l^2}{a^2b^2}}. \] Deduce that if \(b>l>b^2/a\), the position of equilibrium in which the rod is horizontal and above the centre is stable.
Shew that the equation to the envelope of the family of curves \(u+\lambda v+\lambda^2 w=0\), where \(u,v\) and \(w\) are functions of \(x\) and \(y\), is \(4uw=v^2\). Assuming the energy of explosion to be always the same and equal to \(E\), shew that for a gun mounted on a truck and firing a shell of mass \(m\) in a vertical plane parallel to the rails, the horizontal range for an angle of elevation \(\alpha\) of the gun is \[ 4EM/mg\{(M+m)\tan\alpha + M\cot\alpha\}, \] where \(M\) is the mass of the gun and truck together. Shew that the envelope of the trajectories which start from a given point is a parabola with its axis vertical and its focus vertically above the starting point.
A simple pendulum of length \(l\) makes oscillations of angular extent \(\alpha\) on each side of the vertical: find the equation expressing \(d\theta/dt\) in terms of \(\theta\), the inclination of the string to the vertical at time \(t\). If \(\sin\phi = \sin\frac{1}{2}\theta/\sin\frac{1}{2}\alpha\), shew that the period of a complete swing (to and fro) is equal to \[ 4\sqrt{\frac{l}{g}} \int_0^{\pi/2} \frac{d\phi}{\sqrt{(1 - \sin^2\frac{1}{2}\alpha \sin^2\phi)}}. \] The pendulum of a clock is calculated to have a period of 1 second for very small oscillations; shew that if the pendulum is kept swinging through an angle of 8° (so that \(\alpha=4^\circ\)) the clock will lose about 26 seconds a day.
A particle oscillates on a smooth cycloid from rest at a cusp, the axis being vertical and the vertex downwards. Shew that
Eliminate \(x, y, x', y'\) from the following equations: \[ \frac{x}{a} + \frac{y}{b} = 1, \quad \frac{x'}{a'} + \frac{y'}{b'} = 1, \quad x^2+y^2=c^2, \quad x'^2+y'^2=c'^2, \quad xy' - x'y=0. \]
Shew graphically the change in the value of the function \[ (x-a)(x-b)/(x-c)(x-d), \] as \(x\) changes from \(-\infty\) to \(+\infty\), where \(a, b, c, d\) are real numbers such that
Find rationalising factors for the expressions
Solution:
Find the general term of the recurring series whose scale of relation is \[ u_n - u_{n-1} - 5u_{n-2} - 3u_{n-3} = 0, \] and whose first three terms are 2, 5, 20.