Problems

Two conics \(S_1\) and \(S_2\) meet in four distinct points \(A, B, C, D\), and \(O\) is a point on the line \(AB\). The polar of \(O\) with respect to \(S_1\) meets \(S_1\) in \(X, Y\). The lines joining \(C\) to \(X, Y\) meet \(S_2\) again in \(P, Q\) respectively. Show that \(PQ\) and \(AB\) are conjugate with respect to the conic \(S_2\).

Show that two circles, in different planes, which have two points common lie on a sphere. A tetrahedron is such that the pairs of opposite edges are perpendicular. Show that the nine-point-circles of each of the four triangles which are formed by the edges of the tetrahedron lie on a sphere.

Find the equation of the circle of curvature at the origin of the parabola whose equation in rectangular Cartesian coordinates is \(y^2=4ax\). Show that the locus of the middle points of chords of the parabola which touch the circle is given by the equation \[ y^2(y^2-4ax) + 4a^2(x^2+y^2-4ax) = 0. \]

If \(lx+my+1=0\) is the equation of a straight line referred to rectangular Cartesian axes, and if \(l^2 + 2\theta lm - m^2 = 1\), find equations to determine the foci of the conic which the line touches, and show that for all values of \(\theta\) the foci lie on the rectangular hyperbola \(x^2 - y^2 = 2\).

Prove that, if \(a, b\) are positive and \(\sqrt{2} > \theta > 1\), the ellipse \(x^2/a^2+y^2/b^2=1\) meets the rectangular hyperbola \(2xy=(\theta^2-1)ab\) in four real points, two of which, \(P, Q\), are in the positive quadrant. Show that the equation of the circle on \(PQ\) as diameter is \[ x^2+y^2 - \theta ax - \theta by + \frac{1}{2}(a^2+b^2)(\theta^2-1) = 0. \] Show that, as \(\theta\) varies, this circle touches a fixed ellipse which has double contact with the director circle of the given ellipse.

The homogeneous coordinates of a point on a conic \(S\) are expressed in the parametric form \((\theta^2, \theta, 1)\). Find the pole \(P\) of the line joining the points \(A(\alpha^2, \alpha, 1)\) and \(B(\beta^2, \beta, 1)\). Show that a conic \(S'\) passes through the vertices of the triangle of reference and the points \(A, B, P\). If \(A\) and \(B\) vary in such a way that the line \(AB\) passes through the fixed point \((\xi, \eta, \zeta)\), show that the conic \(S'\) passes through the fixed point \((1/\zeta, 1/\eta, 1/\xi)\).

Points \(D, E, F\) are taken in the sides \(YZ, ZX, XY\) respectively of a triangle \(XYZ\), so that \(XD, YE, ZF\) are concurrent. A conic \(S_1\) is drawn through \(X, E, F\) touching \(YZ\) at \(D\), and conics \(S_2, S_3\) are defined similarly by cyclic interchange of letters. Show that the fourth point \(P\) of intersection (other than \(D, E, F\)) of the conics \(S_2, S_3\) lies on \(XD\), and that, if \(Q, R\) are defined similarly, then \(EF, QR, YZ\) are concurrent.

A system of \(n\) forces acts in the plane \(xOy\) at the points \((x_1,y_1), (x_2,y_2), \dots, (x_n,y_n)\); the components of the forces along the axes (which are orthogonal) are \[ (X_1,Y_1), (X_2,Y_2), \dots, (X_n,Y_n). \] Shew that the system can be reduced to two forces, one acting at a given point \(P\) and the other acting along a given straight line \(l\). Find these forces if \(P\) is the point \((h,k)\) and the equation to \(l\) is \(ax+by+c=0\).

A uniform ladder \(AB\) of length \(2l\) rests with one end \(A\) on the ground and the other end \(B\) in contact with a smooth vertical wall; the vertical plane through \(AB\) is perpendicular to the plane of the wall. If \(\mu\) is the coefficient of friction between the ladder and the ground, and \(\alpha\) the inclination of \(AB\) to the vertical, prove that \(\mu\) must exceed \(\frac{1}{2}\tan\alpha\). A force just large enough to move the ladder is now applied to the point of the ladder at a distance \(c\) from \(A\); the force acts away from, and perpendicular to, the wall. Find in what conditions the ladder (1) will rotate about \(A\), (2) will slip at \(A\) and \(B\).

A rectangular trap-door of weight \(W\) is free to rotate about two fixed smooth hinges attached to one side; the line joining the hinges makes an angle \(\alpha\) with the vertical. The door is kept in equilibrium in a position such that its plane makes an angle \(\beta\) with the vertical plane \(P\) through the hinges by a force \(F\) applied to the centre of gravity of the door in a direction perpendicular to the plane \(P\). Shew that \[ F = W \sin\alpha \tan\beta, \] and interpret this result if \(\beta \ge \pi/2\).