The lines drawn from two vertices \(A, D\) of a tetrahedron \(ABCD\) perpendicular to the opposite faces meet in a point \(O\). Prove that
Prove that, if a chord \(QQ'\) of a conic whose focus is \(S\) meets the corresponding directrix in \(Z\), then \(SZ\) is one of the bisectors of the angle \(QSQ'\). Prove the following construction for finding the points in which the conic cuts a given line: Let the given line cut the directrix in \(Z\) and the tangent at the vertex \(A\) in \(T\); from \(T\) draw a line parallel to \(ZS\) cutting the circle whose centre is \(S\) and radius \(SA\) in \(P, P'\). Then the points in which \(SP, SP'\) cut \(ZT\) are the points in which the conic cuts the given line.
Prove that, if the normal at \(P\) to a conic whose focus is \(S\) meets the axis in \(G\), then \(SG:SP\) is constant. Prove also that, if \(PSP'\) is a focal chord, the line joining the point of intersection of the normals at \(P\) and \(P'\) to the mid-point of \(PP'\) is parallel to the axis of the conic.
Prove that \(a+b+c+d\) is a factor of the expression \[ (a+c)(a+d)(b+c)(b+d)-(ab-cd)^2, \] and find the other factor.
If \[ (1+px+x^2)^n = 1+a_1 x + a_2 x^2 + \dots + a_{2n}x^{2n}, \] prove that \[ a_r = a_{2n-r}. \] Shew that \[ 1+3a_1+5a_2+\dots+(4n+1)a_{2n} = (2n+1)(2+p)^n. \]
Expand \(\log(1-x-x^2)\) as far as the term containing \(x^5\), and if \[ \log(1-x-x^2) = -u_1 x - \frac{1}{2}u_2 x^2 - \frac{1}{3}u_3 x^3 - \dots, \] obtain an expression for \(u_n\), and prove that \[ u_n = u_{n-1} + u_{n-2}. \]
\(AB, BC, DE\) and \(EF\) are four equal rods; a hinge at \(B\) connects \(AB\) and \(BC\), and a hinge at \(E\) connects \(DE\) and \(EF\); also \(AB\) and \(DE\) are hinged together at their middle points and \(BC\) and \(EF\) are hinged together at their middle points. Equal forces, applied at \(A\) and \(D\) in directions perpendicular to \(AB\) and \(DE\) are balanced by forces applied at \(C\) and \(F\). Shew that these forces must be equal and their directions must make angles \(\theta\) with \(CF\), such that \(\cot\theta=4\tan\alpha+\cot\alpha\), where \(2\alpha\) is the angle between the rods \(AB\) and \(DE\).
Show how the self-corresponding points of two co-basal homographic ranges may be determined. Give a construction for the points, if any, in which a conic through five given points meets a given straight line.
Prove that the director circles of the conics inscribed in a given quadrilateral form a co-axal system, the radical axis of which is the directrix of the parabola inscribed in the quadrilateral.
Find an equation for the lengths of the axes of the section of the quadric \[ ax^2+by^2+cz^2+2fyz+2gzx+2hxy=1 \] by an arbitrary plane \[ lx+my+nz=0, \] and obtain the condition for a circular section.