Any two perpendicular diameters \(POP'\), \(QOQ'\) of an ellipse are drawn; shew that the four lines \(PQ, PQ', P'Q, P'Q'\) touch a fixed circle with centre \(O\). Generalise the theorem by projection, and deduce (or prove otherwise) the following theorem: Any two chords \(POP'\), \(QOQ'\) of a conic are drawn through a fixed point \(O\), so as to be harmonically conjugate with respect to two fixed lines \(OA, OB\). Prove that the lines \(PQ, PQ', P'Q, P'Q'\) touch a fixed conic, which touches \(OA, OB\) at the points \(A, B\) where they meet the polar of \(O\) with respect to the given conic.
Shew that if two points on a bar are constrained to move along two perpendicular straight lines, the locus of any other marked point on the bar is an ellipse of which the two given lines are the principal axes.
If \(\lambda_1, \lambda_2\) are the roots of the equation in \(\lambda\), \[ \begin{vmatrix} a-\lambda, & b \\ c, & d-\lambda \end{vmatrix} = 0, \] verify that \(\alpha_1=b/(\lambda_1-a), \alpha_2=b/(\lambda_2-a)\) are the roots of the equation in \(x\), \[ cx^2+(d-a)x-b = 0. \] Shew that the equation \(y = (ax+b)/(cx+d)\) can be written in the form \[ \frac{y-\alpha_1}{y-\alpha_2} = \frac{\lambda_2}{\lambda_1} \left(\frac{x-\alpha_1}{x-\alpha_2}\right), \] except when \(\lambda_2 = \lambda_1\); and that in this exceptional case \[ \frac{1}{y-\alpha_1} = \frac{1}{x-\alpha_1} + \frac{c}{\lambda_1}. \]
Give a discussion of the method of ``proportional parts'' as applied to interpolation in mathematical tables; and by considering the function \[ F(x) = f(x) - f(a) - \frac{x-a}{b-a}\{f(b)-f(a)\} - C(x-a)(x-b), \text{ or otherwise,} \] shew that the error in the value of \(f(c)\) as calculated from the tabular values given for \(x=a, x=b\), is equal to \[ \frac{1}{2}(b-c)(c-a)f''(\gamma) \] in excess of the true value, where \(c\) and \(\gamma\) lie between \(a\) and \(b\). Hence or otherwise determine whether the method can be applied safely to the four figure tables supplied, in the following cases:
Starting from the equations \[ dx = \rho d\phi \cos\phi, \quad dy = \rho d\phi \sin\phi, \] shew that the expansions of the coordinates of a point \(P\) on a curve in powers of \(\phi\), the inclination of the tangent at \(P\) to the tangent at \(O\), are given by \[ x = \rho_0 \phi - \frac{1}{2}\rho_1\phi^2 + \frac{1}{6}(\rho_2 - \rho_0)\phi^3, \quad y = \frac{1}{2}\rho_0\phi^2 + \frac{1}{3}\rho_1\phi^3, \] where terms containing \(\phi^4\) are neglected, and \(\rho_0, \rho_1, \rho_2\) are the values of \(\rho, d\rho/d\phi\) and \(d^2\rho/d\phi^2\) at \(O\). Expand \(x \cot\phi + y\) in powers of \(\phi\) as far as the term in \(\phi^2\). The normals at \(O\) and \(P\) meet in \(N\); shew that in general \((ON - \rho_0)\) is equal to \(\frac{1}{2}(\rho - \rho_0)\) and is of order \(\phi\); but that when \(\rho_1=0\), the value is \(\frac{1}{4}(\rho - \rho_0)\), and is of order \(\phi^2\).
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.