Shew that four normals to an ellipse can be drawn through a general point of its plane. Shew that the normals at three points whose eccentric angles are \(\theta_1, \theta_2, \theta_3\) meet if \[ \sin(\theta_2+\theta_3) + \sin(\theta_3+\theta_1) + \sin(\theta_1+\theta_2) = 0, \] and that the fourth normal through their common point is the normal at the point whose eccentric angle is \(\theta_4\), where \(\theta_1+\theta_2+\theta_3+\theta_4\) is an odd multiple of \(\pi\).
If \(r\) denotes the distance of a point \(Q\) lying on a given curve from a fixed point \(S\) in the plane of the curve, and \(p\) is the perpendicular distance from \(S\) to the tangent at \(Q\) to the given curve, shew that the radius of curvature at \(Q\) is \(r\dfrac{dr}{dp}\). If the given curve is an ellipse of semi-axes \(a\) and \(b\) (\(a>b\)) and \(S\) is a focus, shew that \[ \frac{l}{p^2} = \frac{2}{r} - \frac{1}{a}, \quad \text{where } l=b^2/a, \] and hence determine its maximum and minimum radii of curvature.
Sketch the curve \[ x^3 = 3xy^2 + a^2x + y^2. \] Trace the inverse of the curve in the circle \[ x^2+y^2=1, \] and find the area of a loop of this inverse.
Explain what is meant by saying that a certain event has probability \(r\) (\(0 \le r \le 1\)). \(X\) and \(Y\) are partners at bridge against \(A\) and \(B\). \(X\) is dummy and when he puts his hand on the table \(Y\) sees that six trumps are held by the opponents. Shew that the probability that \(A\) and \(B\) each hold three trumps is \(\dfrac{286}{805}\).
Solution: We are interested whether \(A\) has 3 out of 6 trumps and 10 out of 20 non-trumps. This can be done in \(\binom{6}{3} \cdot \binom{20}{10}\) ways, so the probability is \begin{align*} && p &= \frac{\binom{6}{3} \cdot \binom{20}{10}}{\binom{26}{13}} \\ &&&= 20 \cdot \frac{20!}{26!} \cdot \frac{(13!)^2}{(10!)^2} \\ &&&= 2\frac{13 \cdot 11}{5\cdot 23 \cdot 7} \\ &&&= \frac{286}{805} \end{align*}
Functions \(u(x), v(x)\) are defined by the equations \begin{align*} u''+u=0, &\quad v''+v=0, \\ u(0)=0, &\quad u'(0)=1; \\ v(0)=1, &\quad v'(0)=0, \end{align*} where and \(u'=\dfrac{du}{dx}\), etc. Without using trigonometrical functions, prove that
If \(f(x)\) is a function defined in the interval \((a
A chain consists of two portions \(AC, CB\), each of length \(l\), and of uniform densities \(w, w'\) respectively. \(A\) and \(B\) are attached to two points and the chain hangs under gravity in such a way that \(C\) is at the lowest point. Prove that, if the heights of \(A\) and \(B\) above \(C\) are \(h\) and \(h'\), \[ w\left(h - \frac{l^2}{h}\right) = w'\left(h' - \frac{l^2}{h'}\right). \] Prove also that the curvature at \(C\) of the portions \(AC, CB\) are in the ratio \(w:w'\).
Two equal, perfectly elastic, smooth spheres are suspended by vertical strings so that they are in contact, with their centres at the same level. A third equal sphere falls vertically and strikes the other two spheres symmetrically with velocity \(u\), in such a way that at the moment of impact the three centres are in a vertical plane. Shew that the falling sphere rebounds with velocity \(\dfrac{5u}{7}\) and that the other two spheres begin to move outward with velocity \(\dfrac{2\sqrt{3}u}{7}\). Find the impulsive tension in the strings, given that \(w\) is the weight of each sphere.
A reel consists of a cylinder of radius \(r\) and two rims of radius \(R (>r)\). The mass of the reel is \(M\) and its radius of gyration about its axis is \(k\). It is placed on a perfectly rough horizontal table and the thread is drawn out horizontally from beneath it and passes over a light pulley at the edge of the table. The free end is then attached to a mass \(m\) which is allowed to descend under gravity. Shew that the mass \(m\) descends with acceleration \[ \frac{mg(R-r)^2}{M(R^2+k^2)+m(R-r)^2}. \]