Problems

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

1. [(i)] the equation of \(QR\) is \(x/\alpha+y/\beta+z/\gamma=0\),
2. [(ii)] \(QR\) is a tangent to \(S'\),
3. [(iii)] the triangle \(PQR\) is self-polar with respect to the conic \(x^2/f+y^2/g+z^2/h=0\).

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.

\(A\) and \(B\) are two points at the same level, and \(4a\) apart. \(AC, BD\) are two equal uniform rods of length \(a\sqrt{2}\), free to turn about \(A\) and \(B\). \(C\) and \(D\) are \(2a\) apart, and at a depth \(a\) below \(AB\), being joined by a uniform chain of weight \(W\), which rests in equilibrium with its middle point at a small depth \(\frac{1}{16}a\) below \(CD\). Prove that the weight of each rod is \(7W\) approximately.

Two smooth and perfectly elastic spheres of equal radii, but of masses 1 lb. and 4 lb. respectively, are at rest on a smooth horizontal table. The heavier sphere is projected horizontally and strikes the lighter sphere. Prove that the greatest angle through which the direction of motion of the heavier sphere can be deflected by the impact is \(\sin^{-1}\frac{1}{4}\).

A rigid body consisting of two equal masses joined by a weightless rod rests on a smooth horizontal table. One of the masses receives a horizontal blow perpendicular to the rod. Prove that each mass describes a cycloid. \par If the body is thrown up in the air in any manner, and air resistance is neglected, describe the motion in general terms.