Problems

\(AB, BC\) are two uniform heavy rods of equal length and weight \(W\). The rod \(AB\) can move freely about \(A\), there is a hinge at \(B\), and at \(C\) there is a ring which can move freely along a fixed rod through \(A\) which is inclined downwards at an angle \(\alpha\) to the horizontal. Shew that in equilibrium \(\tan BAC = \frac{1}{2}\cot\alpha\), and that the horizontal component of the action at \(B\) is \(\frac{1}{2} W \sin 2\alpha\).

\(A, B, C, D\) are four points in a plane, and \(A', B', C', D'\) are the circumcentres of the triangles \(BCD, ACD, ABD, ABC\) respectively. Prove that a conic having \(D\) and \(D'\) as foci will touch \(B'C', C'A', A'B'\).

Given a straight line \(AB\) divided into two segments by a point \(P\) shew that the locus of points at which these two segments subtend equal angles is a circle. If a line through \(B\) meets this circle in \(C\) and \(D\) shew that \(CA\) and \(DA\) are equally inclined to \(AB\).

Give an account of the properties of the system of confocal conics \[ \frac{x^2}{a^2+\lambda} + \frac{y^2}{b^2+\lambda} = 1. \] Include in your account a proof that the locus of the poles of a given line \(p\) in regard to the confocals is a straight line perpendicular to \(p\) and meeting it in the point where \(p\) touches a confocal; also that the envelope of the polars of a given point \(O\) in regard to the confocals is a parabola which touches the axes of the confocals and whose directrix is the line joining \(O\) to the centre of the confocals. Taking the general conic (in rectangular coordinates) \[ S \equiv ax^2+2hxy+by^2+2gx+2fy+c=0 \] whose tangential equation is \[ \Sigma \equiv Al^2+2Hlm+Bm^2+2Gnl+2Fmn+Cn^2=0 \] shew that the tangential equation of any conic confocal with \(S\) is of the form \[ \Sigma + \mu(l^2+m^2)=0; \] and hence that the conics confocal with \(S\) are given by \[ \Delta S + \mu D + \mu^2 = 0 \] where \(\Delta\) is the discriminant of \(S\) and \(D=0\) is the equation of its director circle.

A uniform heavy rod of length \(2l\) rests with its ends on a fixed smooth parabola with axis vertical and vertex downwards (latus rectum \(= 4a\)). Shew that if \(l > 2a\) there are three positions of equilibrium and that the horizontal position is then unstable, but that if \(l < 2a\) the only position of equilibrium is horizontal.

(i) \(AOA', BOB'\) are two chords of a conic, and \(P, Q\) are two points on a line through \(O\). Shew that, if \(AP\) and \(BQ\) meet on the conic, \(B'P\) and \(A'Q\) will do the same. (ii) \(A, B, C\) are three points on a given conic and \(O\) is a point on a given line. \(AO, BO, CO\) meet the conic again in \(A', B', C'\), and \(BC, CA, AB\) meet the line in \(A'', B'', C''\) respectively. Shew that the lines \(A'A'', B'B'', C'C''\) meet in a point that lies on the conic, and that, if any conic is drawn through \(A, B, C, O\), its two remaining intersections with the line and the conic are collinear with this point.

Express \[ \frac{57x^3 - 25x^2 + 9x - 1}{(x-1)^2(2x-1)(5x-1)} \] as a sum of partial fractions; and expand in ascending powers of \(x\) as far as the term in \(x^4\).

Define a determinant of any order; and give an account, with proofs as far as you think desirable, of the chief properties of determinants of order not exceeding three, including their application to the solution of simultaneous linear equations. By multiplying together the two determinants, \[ \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix}, \quad \begin{vmatrix} bc-f^2 & fg-ch & hf-bg \\ fg-ch & ca-g^2 & gh-af \\ hf-bg & gh-af & ab-h^2 \end{vmatrix} \] or otherwise, prove that the second is the square of the first.

A chain hangs freely in the form of an arc of a circle. Shew that its weight per unit length at any point varies as the square of the secant of the angle which the radius to that point makes with the vertical.

On each of a system of confocal ellipses the points whose eccentric angles are \(\alpha\) and \(\beta\) are taken. Prove that the locus of the intersection of the tangents at these points is a hyperbola of eccentricity \(\sec \frac{1}{2}(\alpha+\beta)\). Prove also that the lines which join the two points on any ellipse are normals to another hyperbola of the same eccentricity.