Prove that, if opposite edges of a tetrahedron are equal, the line joining the mid-points of any pair of opposite edges is perpendicular to them. Prove that, provided \(ABC\) is an acute-angled triangle, it is possible to construct a point \(D\) such that the tetrahedron \(ABCD\) has its opposite edges equal.
Prove that, if the circle drawn with centre \(O\) and passing through the focus \(S\) of a parabola cuts the directrix in \(M\) and \(N\), the lines through \(O\) that are perpendicular to \(SM\) and \(SN\) are tangents to the parabola. Hence, or otherwise, prove that, if \(P\) and \(Q\) are the points of contact of the tangents drawn to the parabola from \(O\), the triangles \(SOP\) and \(SQO\) are similar.
Prove that, if a parallelogram is circumscribed to an ellipse, its diagonals are conjugate diameters of the ellipse. Prove that, if a pair of conjugate diameters of an ellipse meet the director circle in \(P\) and \(Q\), the line \(PQ\) touches the ellipse.
Solve the equations \begin{align*} (y-c)(z-b) &= a^2, \\ (z-a)(x-c) &= b^2, \\ (x-b)(y-a) &= c^2. \end{align*}
Express as partial fractions \(\displaystyle\frac{ay}{(y+a)^2(y-a)}\) and deduce the partial fractions for \(\displaystyle\frac{x(x^2-1)}{(x^2+x-1)^2(x^2-x-1)}\).
Solution: \begin{align*} && \frac{ay}{(y+a)^2(y-a)} &= \frac{A}{y+a} + \frac{B}{(y+a)^2}+\frac{C}{y-a} \\ \Rightarrow && ay &= A(y+a)(y-a)+B(y-a)+C(y+a)^2 \\ y=a: && a^2 &= 4Ca^2 \\ \Rightarrow && C &= \frac14 \\ y =-a: && -a^2&=-2Ba \\ \Rightarrow && B &= \frac{a}{2} \\ y = 0: && 0 &= -a^2A-aB+Ca^2 \\ && 0 &= -a^2A-\frac{a^2}{2}+\frac{a^2}{4} \\ \Rightarrow && A &= -\frac14 \\ \Rightarrow && \frac{ay}{(y+a)^2(y-a)} &= -\frac{1}{4(y+a)} + \frac{a}{2(y+a)^2}+\frac{1}{4(y-a)} \end{align*} If \(a = x\), and \(y = x^2-1\) then \begin{align*} && \frac{x(x^2-1)}{(x^2+x-1)^2(x^2-x-1)} &= \frac{x}{2(x^2+x-1)^2} + \frac{1}{4(x^2-x-1)} - \frac{1}{4(x^2+x-1)} \end{align*}
Two figures \(ABC..., A'B'C'...\) in the same plane are related in such a way that points correspond to points and straight lines to straight lines. If \(AA', BB', CC'\)... all pass through a fixed point \(O\) show that the meets of \(AB, A'B'; AC, A'C'; BC, B'C';...\) all lie on a fixed straight line. A circle is given in the plane of the paper. Show how to obtain from it, by means of a ruler and pencil only, a conic section and establish criteria to determine whether this conic section is an ellipse, parabola or hyperbola.
Obtain the formulae of transformation from trilinear co-ordinates \(\alpha,\beta,\gamma\) referred to a given triangle to cartesian co-ordinates \(x,y\) referred to given rectangular axes in its plane. By this means find the condition that the lines \[ l\alpha+m\beta+n\gamma=0, \quad \lambda\alpha+\mu\beta+\nu\gamma=0 \] may be perpendicular. Deduce the equation of the director circle of the conic \[ p\alpha^2+q\beta^2+r\gamma^2=0. \]
The equations of two circles in space are \begin{align*} 2x+2y-z=0, &\quad 5x^2+5y^2+8z^2-12yz+12zx-8xy=9, \\ 2x-y+2z=0, &\quad 5x^2+8y^2+5z^2+4yz-4zx-4xy-6x+12y+12z=0. \end{align*} Find whether these circles interlace or not.
Prove that the curvature \(\kappa\) of a twisted curve is given by \[ \kappa^2ds^4 = A^2+B^2+C^2, \] where \[ A = d^2ydz-d^2zdy, \quad B=d^2zdx-d^2xdz, \quad C=d^2xdy-d^2ydx. \] In what sense is this equation invariant? Find the curvature of the curve whose equations are \[ x=a\cos\theta, \quad y=b\sin\theta, \quad z=c\theta. \]
Explain what is meant by the differential \(du\) of a function \[ u=f(x,y,z). \] Account for the identities, with the usual notation, \[ dx=\Delta x, \quad d\Delta x=0, \] the variables \(x,y,z\) being independent. Prove that, if \(f\) is differentiable and \(x,y,z\) are themselves differentiable functions of any number of given variables, then \[ du=Adx+Bdy+Cdz, \] the coefficients \(A,B,C\) being identical with those which occur in the expression for \(du\) when the variables \(x,y,z\) are the original independent variables.