Problems

Explain a general method of finding the Highest Common Factor of two polynomials \(f(x), \phi(x)\). Shew how this method may be used to find polynomials \(F(x), \Phi(x)\), such that \(f(x).F(x) + \phi(x).\Phi(x) = 1\), when \(f(x)\) and \(\phi(x)\) have no common algebraic factor.

Taking \((1/u, \theta)\) as the polar coordinates of a point of a plane curve, obtain an expression for the curvature in terms of \(u, \frac{du}{d\theta}, \text{and } \frac{d^2u}{d\theta^2}\). Shew that the curvature of the curve \(au = \cosh n\theta\) has a stationary value provided \(3n^2\) is not less than 1. Determine whether this value is a maximum or minimum.

Define the envelope of a family of plane curves. If circles are described on focal chords of a parabola as diameters, prove that their envelope consists of the directrix and a circle. Prove also that the points of contact of any circle of the system with the envelope are collinear with the vertex of the parabola.

If \(u=x+y\) and \(v=x-y\), express \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) in terms of \(\frac{\partial f}{\partial u}\) and \(\frac{\partial f}{\partial v}\) where \(f\) is any function of the variables considered. Deduce that the necessary and sufficient condition that \(f\) should be expressible as a function of \(x+y\) only is that \(\frac{\partial f}{\partial x} = \frac{\partial f}{\partial y}\). If \begin{align*} f_0(x) &= 1+\frac{x^4}{4}+\frac{x^8}{8}+\dots, \\ f_1(x) &= x+\frac{x^5}{5}+\frac{x^9}{9}+\dots, \\ f_2(x) &= \frac{x^2}{2}+\frac{x^6}{6}+\frac{x^{10}}{10}+\dots, \\ f_3(x) &= \frac{x^3}{3}+\frac{x^7}{7}+\frac{x^{11}}{11}+\dots, \end{align*} prove that \[ f_0(x)f_3(y)+f_1(x)f_2(y)+f_2(x)f_1(y)+f_3(x)f_0(y) = f_3(x+y). \]

If \(I(r,s) = \int_a^\infty \frac{(x-a)^s}{x^r}dx\), \(s>0, r>s+2\), express \(I(r,s)\) in terms of (a) \(I(r,s-1)\), (b) \(I(r-1,s)\). Assuming \(r\) and \(s\) to be integers satisfying the above inequalities, find the value of \(r-1 I(r,s)\).

If the coordinates \((x,y)\) of any point on a plane curve are expressed as functions of a parameter \(\theta\), interpret the expression \(\frac{1}{2}\int \left(x\frac{dy}{d\theta}-y\frac{dx}{d\theta}\right)d\theta\). Sketch the curve given by \(x=2a(\sin^3\theta+\cos^3\theta)\), \(y=2b(\sin^3\theta-\cos^3\theta)\), and prove that its area is \(3\pi ab\).

Prove that the length of the arc of the curve whose pedal \((p,r)\) equation is \(p=r-d\) between the points \(r=a, r=2a\) is \(a(\sqrt{3}-\frac{\pi}{3})\). Shew that the polar equation of this curve may be written in the form \[ 2r = a\sec^2\left(\frac{\sqrt{2ar-a^2}+\frac{a\pi}{2}-a\theta-a}{2a}\right). \]

If \(\alpha, \beta\) are two of the roots of the cubic equation \(x^3 + 3qx + r = 0\), prove that \(\alpha/\beta\) is one of the roots of the sextic equation \[ r^2(x^2+x+1)^3 + 27q^3x^2(x+1)^2 = 0. \] Explain this result in relation to the roots of the cubic equation, when (i) \(q=0\), (ii) \(r=0\), (iii) \(4q^3+r^2=0\).

Shew that by a suitable choice of the triangle of reference the homogeneous co-ordinates \((x, y, z)\) of any four points may be taken as \((f, \pm g, \pm h)\) and that the locus of the poles of the line \(lx+my+nz=0\) with respect to conics through these four points is given by \(f^2l yz + g^2m zx + h^2n xy = 0\). Explain this result when \(l=0\). Shew also that the locus of centres of conics through four fixed points is a parabola, if and only if, one of these points is at infinity.

The rectangular Cartesian coordinates \((x,y)\) of a point are given by \(x=at^2+2pt\), \(y=bt^2+2qt\), where \(a, b, p, q\) are constants and \(t\) is a parameter; prove that the locus of the point is a parabola, and that this parabola touches the line \(lx+my+1=0\), if \[ (pl+qm)^2 - (al+bm) = 0. \] Find (not necessarily in this order) the equations of the tangent at the vertex, of the axis and of the directrix, and the coordinates of the focus of this parabola. Verify from your results that the length of the latus rectum is \(4(aq-bp)^2/(a^2+b^2)^{3/2}\).