Problems

Write a short account of the method of reciprocation showing particularly how to reciprocate a circle into a conic of any species. Apply the method to the following case: \(S\) is the focus of a given conic and a line \(L\) meets the corresponding directrix in \(Z\). \(L'\) is the line joining \(Z\) to the pole of \(L\). A second conic having a focus at \(S\) touches \(L, L'\). A common tangent to the conic touches them at \(Q, Q'\). Show that \(QSQ'\) is a right angle.

In the continued fraction \(\displaystyle\frac{1}{a_1+}\frac{1}{a_2+}\dots\), the \(n\)th convergent is denoted by \(p_n/q_n\). Prove that

1. [(1)] \(p_{n-1}q_n-q_{n-1}p_n=(-1)^n\),
2. [(2)] \(\displaystyle\frac{p_n}{q_n} = \frac{1}{q_1} - \frac{1}{q_1q_2} + \frac{1}{q_2q_3} - \dots\) to \(n\) terms,
3. [(3)] \(p_{n-1}+q_n\) is not altered, if the numbers \(a_1, a_2, \dots, a_n\) are permuted cyclically.

Prove that if \(A,P,Q\) are polynomials in \(x\) and \(A\) is of lower degree than \(PQ\), then \(A/PQ\) can be expressed in the form \(M/P+N/Q\), where \(M,N\) are respectively of lower degrees than \(P,Q\). Expand \(\displaystyle\frac{\sin\phi}{1-2x\cos\phi+x^2}\) in ascending powers of \(x\), and prove that the remainder after \(n\) terms is equal to \[ \frac{x^n\sin(n+1)\phi-x^{n+1}\sin n\phi}{1-2x\cos\phi+x^2}. \]

Determine the different kinds of conics represented by the equation \[ x^2+4\lambda xy+4y^2+2(1+\lambda)x+8y+5+2\lambda=0, \] for different values of \(\lambda\). Examine especially the critical cases \(\lambda=1, 0, -1, -2\) and illustrate by sketches the transition from one kind of conic to another.

Show that the coordinates of any point on a conic can be expressed in terms of a parameter by the equations \[ \frac{x}{at^2+2bt+c} = \frac{y}{a't^2+2b't+c'} = \frac{1}{a''t^2+2b''t+c''}. \] Find the condition that \(lx+my+n=0\) may be a tangent, and obtain (i) the foci, (ii) the director circle, (iii) the conditions for the conic to be a parabola, or a rectangular hyperbola.

Show that the function \(\sin x+a\sin 3x\) for values of \(x\) between \(0\) and \(\pi\) has two minima with an intermediate maximum if \(a<-\frac{1}{9}\); one maximum if \(-\frac{1}{9}\frac{1}{9}\).

Give an account of some method of finding the rectilinear asymptotes of a curve whose \(x,y\) equation is given and show how to determine in what manner the curve approaches an asymptote. Consider the cases

1. [(i)] \(y(x^2-1)=x^3+x^2+1\),
2. [(ii)] \(x(y-x)^2-3y(y-x)+2x=0\).

Give a general account of the motion of a projectile, neglecting air resistance. Consider the possible paths through a given point \(P\) when the velocity at the point of projection \(O\) is given in magnitude, and the envelope of the paths when the direction is varied for a given magnitude. A fort is on the edge of a cliff of height \(h\). Show that there is an annular region in which the fort is out of range of the ship, but the ship is not out of range of the fort, of area \(8\pi kh\), where \(\sqrt{2gk}\) is the velocity of the shells used by both.

A particle on a smooth table is attached to a string passing through a small hole in the table and carries an equal particle hanging vertically. The former particle is projected along the table at right angles to the string with velocity \(\sqrt{2gh}\) when at a distance \(a\) from the hole. If \(r\) is the distance from the hole at time \(t\), prove the results

1. [(i)] \(2\left(\frac{dr}{dt}\right)^2 = 2gh\left(1-\frac{a^2}{r^2}\right)+2g(a-r)\),
2. [(ii)] the lower particle will be pulled up to the hole if the total length of string is less than \(a+h+\sqrt{ah+\frac{1}{2}h^2}\),
3. [(iii)] the tension of the string is \(\frac{1}{2}mg\left(1+\frac{2a^2h}{r^3}\right)\), \(m\) being the mass of each particle.