Problems

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.

If the diagonals of a quadrilateral inscribed in a circle are perpendicular to each other, prove that the length of the perpendicular from the centre of the circle to any side of the quadrilateral is equal to half the opposite side of the quadrilateral.

P is a point on a hyperbola whose foci are S, H and \(SP>HP\); if T and T' are the points of contact of the tangents from H to the circle whose centre is S and radius is \(SP-HP\), prove that ST, ST' are parallel to the asymptotes of the hyperbola. \par Hence, or otherwise, prove that the axes of two parabolas with a common focus F and common points A, B are parallel to the asymptotes of the hyperbola passing through F and having A, B as foci.

``The straight lines which cut two conics S, S' in pairs of points which are harmonically conjugate touch another conic \(\Sigma\).'' \par Assuming this result, prove that, if S, S' are circles with centres at O, O' and with common points P, Q,

1. [(i)] \(\Sigma\) touches the two pairs of tangents to S, S' at P, Q;
2. [(ii)] the foci of \(\Sigma\) are O, O'.

If A, B, C, D are four coplanar points, prove that the three pairs of lines through any point P parallel to the pairs of lines (BC, AD), (CA, BD), (AB, CD) are in involution. \par If the double lines of this involution are perpendicular and the points A, B, C are fixed, find the locus of the fourth point D.

Prove that there are two real points P, Q in space at each of which the sides of a given acute-angled triangle subtend a right angle and that PQ passes through the orthocentre of the triangle.

If \(p_1 = a_1x+b_1y+c_1, \quad p_2 = a_2x+b_2y+c_2, \quad p_3 = a_3x+b_3y+c_3\),

1. [(i)] find the harmonic conjugate of the line \(p_2+\lambda p_3=0\) with respect to the pair of lines \(p_2=0, p_3=0\);
2. [(ii)] prove that the centroid of the triangle, whose sides lie along the lines \(p_1=0, p_2=0, p_3=0\), is given by the equations \[ (a_2b_3-a_3b_2)p_1 = (a_3b_1-a_1b_3)p_2 = (a_1b_2-a_2b_1)p_3. \]