The line \(lx+my+n=0\) intersects the circle \(x^2+y^2+2gx+2fy+c=0\) in \(A\) and \(B\). \(O\) is the origin, and the straight lines \(OA, OB\) intersect the circle again in \(P\) and \(Q\). Shew that the equation of the straight line \(PQ\) is \[ c(lx+my+n)=2n(gx+fy+c). \] As \(n\) varies find the least distance of the point of intersection of \(AB\) and \(PQ\) from the origin, stating the corresponding value of \(n\).
\(ax+by+c=0\) is one asymptote of a hyperbola which passes through the origin and which touches the straight line \(bx+ay=c\) at the point \(\left(\dfrac{c}{2b}, \dfrac{c}{2a}\right)\). Find the equation of the hyperbola and of the normal at the origin. Prove that if this normal is parallel to the other asymptote then \((a-b)^4 = 8a^2b^2\).
Shew that if the general equation of the second degree represents a parabola then the terms of the second degree form a perfect square. Find the equation of the reflection of the parabola whose equation is \[ 16x^2+9y^2+24xy-81x+18y+34=0 \] in the tangent at its vertex.
\(\Delta\) and \(R\) are respectively the area and the circumradius of a triangle \(ABC\); \(\delta\) and \(r\) are respectively the area and the inradius of the pedal triangle of \(ABC\). Prove that \(\dfrac{\delta}{\Delta} = \dfrac{r}{R} = x\), where \(x\) is a root of the equation \[ x^2+4x\cos^2 B\cos^2 C + 2\cos^2 B\cos^2 C(\cos 2B+\cos 2C) = 0. \] Write down the other root of this equation in terms of \(B\) and \(C\).
Obtain the \(n\)th roots of \(a+b\sqrt{-1}\), where \(a\) and \(b\) are real. If \(\omega\) is one of the imaginary \(2n\)th roots of unity, prove that \[ \sin\theta+\omega\sin 2\theta+\omega^2\sin 3\theta+\dots+\omega^{2n-1}\sin 2n\theta = \frac{\sin\theta+\omega\sin 2n\theta - \sin(2n+1)\theta}{1-2\omega\cos\theta+\omega^2}. \] Deduce the sums of the series \[ \sin\theta+\omega^2\sin 3\theta+\omega^4\sin 5\theta+\dots+\omega^{2n-2}\sin(2n-1)\theta \] and \[ \sin 2\theta+\omega^2\sin 4\theta+\omega^4\sin 6\theta+\dots+\omega^{2n-2}\sin 2n\theta, \] and find all the values of \(\theta\) which make these two sums equal.
\(D\) is a point in the base \(BC\) of a triangle \(ABC\), and a line through \(D\) meets \(AB\) and \(AC\) in \(B'\) and \(C'\) respectively. \(D\) divides \(BC\) and \(B'C'\) in the ratios \(\lambda:\mu\) and \(\lambda':\mu'\) respectively. Shew that the areas of the triangles \(ABC\) and \(AB'C'\) are in the ratio \[ \lambda'\mu'(\lambda+\mu)^2 : \lambda\mu(\lambda'+\mu')^2. \]
Shew that the inverse of a circle through the centre of inversion is a straight line. \(AB\) is a chord of a given circle, and \(CD\) is the diameter perpendicular to \(AB\). By inversion with respect to \(A\) shew that all circles which touch the given circle and the line \(AB\) are orthogonal to one or other of two circles with centres at \(C\) and \(D\) respectively, and that the chords of contact pass through \(C\) or \(D\). Examine how the different types of circle touching \(AB\) and the given circle arise from the inverted figure.
\(A\) and \(B\) are two fixed points on a fixed circle. \(PQ\) and \(P'Q'\) are a variable pair of chords parallel to and equidistant from the chord \(AB\). Shew that the lines \(PP', PQ', QP'\) and \(QQ'\) touch a fixed parabola, which is also touched by the tangents to the circle at \(A\) and \(B\). What is the projective generalisation of this theorem?
Explain what is meant by an involution on a conic and shew that the joins of pairs of points of an involution pass through a fixed point. \(A_1, A_2, \dots, A_6\) are given points on a given conic. Shew that the six points \(A_5, A_6, (15, 26), (16, 25), (35, 46)\) and \((36, 45)\) lie on a conic, where \((15, 26)\) for instance denotes the point of intersection of \(A_1A_5\) and \(A_2A_6\). Prove also that the two remaining intersections of the two conics are the points of contact of tangents to the given conic from the point \((12, 34)\).
Two circles, with centres \(A\) and \(B\) and radii \(a\) and \(b\), lie in different planes which meet in a line \(l\). Shew that the two circles will lie on the same sphere provided that \(AB\) is perpendicular to \(l\) and that \[ AP^2 - BP^2 = a^2 - b^2, \] where \(P\) is any point on \(l\). Shew that these conditions are always satisfied if the two circles have two common points.