Problems

Prove that if \[ a\frac{y+z}{y-z} = b\frac{z+x}{z-x} = c\frac{x+y}{x-y}, \] each of these expressions \(= \pm \left\{-\frac{abc}{a+b+c}\right\}^{\frac{1}{2}}\).

Reduce the equation \(x^3+3px^2+3qx+r=0\) to the form \(y^3+3y+m=0\) by assuming \(x=\lambda y + \mu\); and solve this equation by assuming \(y=z-\frac{1}{z}\). Hence prove the condition for equal roots to be \[ 4(p^2-q)^3 = (2p^3-3pq+r)^2. \]

Shew that a uniform flexible chain hangs under gravity in a catenary whose Cartesian equation can be written in the form \[ y = c \cosh \frac{x}{c}. \] If the chain is of length \(l\) and hangs between two points whose horizontal and vertical distances apart are \(a, b\), respectively, prove that the parameter \(c\) of the catenary is given by \[ (l^2-b^2)^{\frac{1}{2}} = 2c\sinh\frac{a}{2c}. \]

Tangents are drawn to the parabola \(y^2=4ax\) at the points whose ordinates are \(2am_1, 2am_2, 2am_3\). Find the equation of the circumcircle of the resulting triangle, and from the equation show that it passes through the focus. Show also that, whatever be the value of \(k\), the same circle circumscribes the triangle formed by the tangents at three points whose ordinates are \(2a\mu_1, 2a\mu_2, 2a\mu_3\), where \(\mu_1, \mu_2, \mu_3\) are the roots of \[ (\mu-m_1)(\mu-m_2)(\mu-m_3)+k(\mu^2+1)=0. \]

Find for what values of \(a\) and \(b\) the roots of the equation \[ x^4 - 4x^3 + ax^2 + bx - 1 = 0 \] are in arithmetical progression.

Trace the curve \(4(x^2+2y^2-2ay)^2=x^2(x^2+2y^2)\) and find the radii of curvature of the two branches at the origin.

A uniform beam \(AB\) of length \(l\) and weight \(w\) per unit length is smoothly hinged at \(A\), and is kept at an inclination of 45\(^\circ\) to the upward vertical through \(A\) by a light horizontal rope which joins a fixed point to the point of the beam distant \(l/3\) from \(A\). Find expressions for the thrust, shearing force and bending moment at any point of the beam.

Show that the locus of the point of intersection of the normals at the pairs of points in which a given straight line meets the conics of a confocal system is a straight line.

Prove that \[ \frac{1}{3\left(1 - \frac{1}{2^2}\right)} - \frac{1}{4\left(1 + \frac{1}{3^2}\right)} + \frac{1}{5\left(1 - \frac{1}{4^2}\right)} - \dots = \frac{1}{2}. \]

Shew that \[ \frac{\frac{\partial^2 z}{\partial x^2}\frac{\partial^2 z}{\partial y^2} - \left(\frac{\partial^2 z}{\partial x \partial y}\right)^2}{\left(\frac{\partial z}{\partial x}\right)^2 \left(\frac{\partial z}{\partial y}\right)^2} = \frac{\frac{\partial^2 x}{\partial y^2}\frac{\partial^2 x}{\partial z^2} - \left(\frac{\partial^2 x}{\partial y \partial z}\right)^2}{\left(\frac{\partial x}{\partial y}\right)^2 \left(\frac{\partial x}{\partial z}\right)^2}, \] where \(x,y,z\) are variables connected by one relation, which is conceived on the left-hand side as determining \(z\) as a function of \(x\) and \(y\), and on the right-hand side \(x\) as a function of \(y\) and \(z\).