Problems

By proving that the Simson Line of a point on the circumcircle of a triangle bisects the join of the point to the orthocentre, or otherwise, shew that the orthocentres of the four triangles formed by four straight lines are collinear, and that the four circumcircles have a common point. \par Apply this result to the case of four tangents to a parabola.

Shew that the inverse of a circle with respect to a point not necessarily in its plane is either a circle or a straight line. \par If \(P_1\) and \(P_2\) are a pair of points inverse with respect to the circle S, shew by inversion with respect to a coplanar point O not on S that they become \(P_1'\) and \(P_2'\), a pair of points inverse with respect to the inverse of S. \par State the result in the cases (i) when O does lie on S, and (ii) when O does not lie in the plane of S.

Prove that a circle through the vertex of a parabola cuts the curve again in three points at which the normals to the parabola are concurrent. \par Taking the co-ordinates of this point of concurrence as \((h,k)\) with the equation of the parabola as \(y^2=4ax\), find the co-ordinates of the centre of the circle.

Shew that the locus of the poles of a fixed straight line with respect to conics through four fixed points is also a conic. \par Apply this result to the case of a system of coaxal circles, and shew that one point of the resulting conic lies on the radical axis where it is cut by the circle orthogonal to the system, and passing through the intersection of the radical axis with the given line.

Taking the equation of a straight line as \(lx+my=1\), shew that the tangential equation \(Hlm+Ul+Vm=0\) represents a parabola. \par Find the co-ordinates of the focus and the equation of the directrix.

A conic, inscribed in a triangle ABC, touches BC, CA, AB, at A', B', C', respectively. Shew that if any other conic through A', B', C' cuts the sides of the triangle again in A'', B'', C'', then AA'', BB'', CC'' are concurrent.

Find the locus of points from which a pair of perpendicular tangents can be drawn to a conic. Discuss any exceptional cases. \par If the equation of a system of confocal conics is taken as \(\dfrac{x^2}{a^2+\lambda}+\dfrac{y^2}{b^2+\lambda}=1\), find the locus of points from which a tangent can be drawn to the conic given by \(\lambda=\lambda_1\) to be perpendicular to a tangent from the same point to the conic given by \(\lambda=\lambda_2\). \par Shew also that from such points perpendicular tangents can be drawn to a third conic of the system given by \(\lambda=\lambda_3\); and find \(\lambda_3\) in terms of \(\lambda_1\) and \(\lambda_2\).

Shew that two non-intersecting straight lines have a mutual perpendicular which is the shortest distance between them. \par \(A_1\) and \(A_2\) are any two points on a straight line \(p\), and \(B_1\) and \(B_2\) are any two points on a straight line \(q\) which does not meet \(p\). \(C_1\) divides \(A_1B_1\) and \(C_2\) divides \(A_2B_2\) in the ratio \(\lambda/\mu\). Shew that the mutual perpendicular of \(C_1C_2\) and the line of shortest distance between \(p\) and \(q\) cuts the latter in the ratio \(\lambda/\mu\). \par If further, \(D_1\) divide \(A_1B_2\), and \(D_2\) divide \(A_2B_1\) in the ratio \(\lambda/\mu\), prove that \(C_1C_2\) and \(D_1D_2\) bisect each other.

Prove that

1. [(i)] \(\dfrac{p!(p+1)!}{(2p+1)!}2^{2p}\cos^{2p+1}\theta = \cos\theta + \dfrac{p}{p+2}\cos 3\theta + \dfrac{p(p-1)}{(p+2)(p+3)}\cos 5\theta - \dots + \dfrac{p!}{(p+2)(p+3)\dots(2p+1)}\cos(2p+1)\theta\),
2. [(ii)] \(\dfrac{p!p!}{2p!}2^{2p-1}\sin^{2p}\theta = \frac{1}{2} - \dfrac{p}{p+1}\cos 2\theta + \dfrac{p(p-1)}{(p+1)(p+2)}\cos 4\theta - \dots + (-1)^p \dfrac{p!}{(p+1)(p+2)\dots 2p}\cos 2p\theta\),

where \(p\) is a positive integer. \par Hence, or otherwise, find the sum of the terms: \[ 1 + \frac{2n!2n!}{(2n-2)!(2n+2)!}\cos 4\theta + \frac{2n!2n!}{(2n-4)!(2n+4)!}\cos 8\theta + \dots + \frac{2n!2n!}{4n!}\cos 4n\theta, \] where \(n\) is a positive integer.

Prove the existence of the nine-point circle for any triangle. \par Shew that the sum of the squares of the distances of the nine-point circle from the vertices and orthocentre of the triangle is three times the square of the circumradius.