Problems

Show that, if \(P, Q, R, S\) are four concyclic points on a conic, the lines \(PQ, RS\) are equally inclined to the principal axes. Deduce that the six intersections of three circles lie on a conic if and only if the centres of the circles are collinear.

\(P\) is any point of the parabola \[ y^2=a(x-a) \] and \(O\) is the vertex of the parabola \[ y^2=4ax. \] The circle on \(OP\) as diameter meets the second parabola in three points other than \(O\). Prove that the normals at these three points meet in a point of the parabola \[ y^2 = 4a(x+a). \]

The lines joining a point \(P\) of a rectangular hyperbola \[ S \equiv xy - c^2 = 0 \] to the points \((a,a), (-a,-a)\) meet \(S\) again in \(Q, R\). Show that the locus of the pole of \(QR\) with respect to \(S\) is the hyperbola \[ (a^2x+c^2y)(c^2x+a^2y) = c^2(c^2+a^2)^2. \]

Chords of a conic \(S\) are drawn subtending a right angle at the fixed point \(P\). Prove that their envelope is a conic with \(P\) as one of its foci, and the polar of \(P\) with respect to \(S\) as the corresponding directrix.

\(S\) is the conic \[ S \equiv ax^2+2hxy+by^2+2gx+2fy+c = 0, \] and \(S'\) is the circle \[ S' \equiv x^2+y^2-r^2=0. \] Find the equation of the locus of the centres of the conics of the pencil \(S+\lambda S'=0\). Show that it meets \(S\) in the feet of the normals drawn from the centre of \(S'\) to \(S\).

The line \(lx+my+nz=0\) cuts the sides \(YZ, ZX, XY\) of the triangle of reference \(XYZ\) in the points \(L, M, N\) respectively. \(L'\) is the harmonic conjugate of \(L\) with respect to \(Y, Z\), and \(M', N'\) are similarly defined. \(O\) is the point \((p,q,r)\). \(OL'\) cuts \(M'N'\) at \(U\), and \(V, W\) are similarly defined. Prove that \(XU, YV, ZW\) are concurrent at the point whose coordinates are given by the equations \[ l(-lp+mq+nr)x = m(lp-mq+nr)y = n(lp+mq-nr)z. \]

Prove that in general there are two points in the plane of three coplanar circles such that the lengths of tangents drawn from either point to the circles is in a given ratio. What happens if the three tangents are of equal length? Shew that there is one fixed circle orthogonal to any circle which passes through a pair of points such that the ratio of the lengths of tangents drawn from either point to three fixed circles is the same, and that this orthogonal circle is independent of the value of the ratio.

Shew that the inverse of a sphere with respect to a centre of inversion on or inside it is a sphere or a plane. Lines are drawn from a given point \(P\) to cut a sphere in variable points \(Q_1\) and \(Q_2\). Shew that if \(Q_1\) lies on a circle, so does \(Q_2\).

Four straight lines in a plane are drawn so that \(AB, CD\) intersect in \(E\), and \(AD, BC\) intersect in \(F\). Prove that the mid-points of \(AC, DB, EF\) are collinear.

Prove that any straight line is cut in pairs of points in involution by conics passing through four fixed points. Hence or otherwise shew that in general through four given points there pass two parabolas and one rectangular hyperbola.