(i) The two sets of points \(P_1, P_2, \dots\) on a line \(OX\), and \(Q_1, Q_2, \dots\) on a line \(OY\) are homographic, \(P_r\) and \(Q_r\) being corresponding points; prove that the intersections of pairs of lines such as \(P_rQ_s\) and \(P_sQ_r\) lie on a line. \par (ii) State the condition that the conic which is the envelope of \(P_rQ_r\) should be a pair of points. \(O, A, B\) are three fixed points on a fixed line, and \(O, P, Q\) are three fixed points on a line which rotates round \(O\) in a plane through the fixed line; prove that the locus of the intersection of \(AP, BQ\) is a circle and identify its centre and its radius.
Explain what is meant by the statement that two pairs of points on a conic are harmonic. \par \(O, X\) are two fixed points on a conic, and \(OP, OQ\) are variable chords of the conic equally inclined to \(OX\); prove that the chord \(PQ\) passes through a fixed point on the tangent at \(X\) to the conic.
Prove that the points of contact of the tangent lines from a point \(P\) to a sphere lie on a plane \(p\) (the polar plane of \(P\)), and that, if \(q\) is the polar plane of \(Q\), then the polar plane of any point on the line \(PQ\) is collinear with \(p, q\). \par Prove also that the lines \(PQ, pq\) are perpendicular to each other and that the feet of their common normal are inverse points with respect to the sphere. \item[] N.B. The equations in Questions 6, 7, 8, 9 are referred to rectangular Cartesian coordinate axes.
Prove that \[ \{(b-b')x - (a-a')y + ab' - a'b\}^2 = \{(r-r')x+ar'-a'r\}^2 + \{(r-r')y+br'-b'r\}^2 \] is the equation of a pair of common tangents to the circles \[ (x-a)^2+(y-b)^2 = r^2, \quad (x-a')^2+(y-b')^2 = r'^2, \] and write down the equation of the other pair of common tangents to these circles.
Prove that the line \(2tx-y=2kt^3+kt\), where \(t\) is a parameter, is a normal to the parabola \(y^2=kx\). \par The normals to this parabola at the points of contact of tangents from \((x_1, y_1)\) meet at \(P\); prove that the normal to the parabola at the point \((4y_1^2/k, -2y_1)\) also passes through \(P\).
The line \(lx+my+n=0\) cuts the conic \(ax^2+by^2+c=0\) at the points \(A, B\) and the circle on \(AB\) as diameter cuts the conic again at the points \(P, Q\); find the equation of the line \(PQ\), and prove that, if \(AB\) is a variable tangent to the conic \(px^2+qy^2+r=0\), then \(PQ\) touches the conic \((a-b)^2(px^2+qy^2)+(a+b)^2r=0\).
The coordinates of any four points \(A, B, C, D\) are taken as \((t, \frac{1}{t})\), where \(t=a,b,c,d\); shew that the coordinates of the circumcentre of the triangle \(ABC\) are \[ \left\{ \frac{1}{2}\left(a+b+c+\frac{1}{abc}\right), \frac{1}{2}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+abc\right) \right\}, \] and that, if \(a^2b^2c^2d^2 \ne 1\), the centre of the rectangular hyperbola through the circumcentres of the triangles \(BCD, CAD, ABD, ABC\) has coordinates \[ \left\{ \frac{1}{2}\left(a+b+c+d\right), \frac{1}{2}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right) \right\}. \]
The homogeneous coordinates of any point \(P\) on the conic \(S \equiv fyz+gzx+hxy=0\) are \((f/\alpha, g/\beta, h/\gamma)\), where \(\alpha, \beta, \gamma\) are parameters, such that \(\alpha+\beta+\gamma=0\); the tangents from \(P\) to the conic \(S' \equiv x^2+y^2+z^2-2yz-2zx-2xy=0\) cut the conic \(S\) again at points \(Q, R\). Prove that
Spheres of weights \(w, w'\) rest on different and differently inclined planes. The highest points of the spheres are connected by a light horizontal string perpendicular to the common horizontal edge of the two planes and above it. If \(\mu, \mu'\) are the coefficients of friction and if each sphere is on the point of slipping down, prove that \(\mu w = \mu'w'\).
A rod \(PQ\) of length \(c\) has its centre of gravity at \(G\), and hangs from a small smooth peg by a light inextensible string of length \(b\), which is attached to the ends of the rod and passes over the peg. If \(\frac{c^2}{b}\) is greater than the difference between \(PG\) and \(QG\), prove that there is a position of equilibrium in which the rod is not vertical.