Problems

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 \[ (a_2b_3-a_3b_2)p_1 = (a_3b_1-a_1b_3)p_2 = (a_1b_2-a_2b_1)p_3. \]

If a circle of radius R cuts a rectangular hyperbola whose centre is O at the points A, B, C, D, prove that \[ OA^2+OB^2+OC^2+OD^2 = 4R^2. \]

The equations of a conic and a line referred to rectangular axes are \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \quad \text{and} \quad lx+my+n=0; \] prove that the line is a principal axis of the conic, if \[ al+hm : hl+bm : gl+fm = l:m:n, \] and deduce that the equation of the principal axes is \[ h(u^2-v^2)+(b-a)uv = 0, \] where \(u \equiv ax+hy+g, v \equiv hx+by+f\). \par Interpret these results geometrically, when \(a=\alpha^2, h=\alpha\beta, b=\beta^2\).

If \(lx+my+1=0\) is the equation of a straight line referred to rectangular axes, interpret geometrically the constants \(a, b, a', b', c\) in relation to the conic whose tangential (envelope) equation is \[ (al+bm+1)(a'l+b'm+1) = c^2(l^2+m^2). \] If \(\Sigma'\) is any conic touching the two pairs of tangents from the points P, Q to a conic \(\Sigma\), prove that the four common tangents of any conic confocal with \(\Sigma\) and of any conic confocal with \(\Sigma'\) touch a conic with foci at P, Q.

If P, Q, R are three points with homogeneous coordinates \((p, g, h), (f, q, h), (f, g, r)\), respectively, and XYZ is the triangle of reference, find the equation of the line through the three intersections of the pairs of lines (QR, YZ), (RP, ZX), (PQ, XY). \par Shew also that, if \(fgh=pqr\), the lines XQ, YR, ZP are concurrent and the lines XR, YP, ZQ are concurrent.