Prove that the general equation of a conic whose centre is the origin and which cuts the lines \(x=a\), \(y=\beta\) at right angles is \[ \frac{x^2}{\alpha^2} + 2\lambda xy + \frac{y^2}{\beta^2} + \lambda\left(1 - \frac{x^2}{\alpha^2} - \frac{y^2}{\beta^2}\right) = 0. \]
A mine cage, weighing with its load 5 cwt., is raised by an engine which exerts a constant turning moment on the rope drum which is 16 ft. in diameter. The speed rises until the engine is running at 60 revolutions per minute, when its output is 55 horse-power. \par Find the acceleration and the time that elapses before the cage reaches full speed: also find how far the cage rises in that time.
Find the equation of the circle which passes through the origin, has its centre on the line \(x+y=0\), and cuts the circle \[ x^2+y^2-4x+2y+4=0 \] at right angles.
Give a discussion of the hodograph and its applications. Shew that the motion of a moving point is completely given when its path and the hodograph of its motion constructed with a given pole are given. Illustrate your arguments by reference to the hodograph of a point on the rim of a wheel rolling with uniform velocity along a level road. \par A point describes a circle of radius \(a\) so that its hodograph is a second circle of radius \(b\). If the pole of the hodograph be at distance \(c\) from its centre, where \(c/b\) is small, shew that the time of a complete revolution is approximately \[ 2\pi a(1+\tfrac{1}{2}c^2/b^2)/b. \]
A dynamo, of E.M.F. 105 volts and internal resistance 0.025 ohm, is in parallel with a storage battery of E.M.F. 100 volts and internal resistance 0.06 ohm. They are feeding an external circuit of resistance 1.75 ohm: find whether the battery is charging or discharging, and calculate the current through the dynamo and the P.D. at the terminals of the external circuit.
Using areal (or trilinear) coordinates, find the coordinates of the centre of a conic circumscribing the triangle of reference. \par Two conics circumscribe a triangle and touch one another at one of the angular points. Prove that their two centres, their point of contact, and the middle points of the sides of the triangle lie upon a conic.
A spring of negligible inertia carries a pan weighing 1 ounce, and is such that a \(\frac{1}{2}\) lb. weight will lower the pan by 1 inch. It is compressed 2 inches and placed on a table with its axis vertical: a 2 ounce weight is put on it and the spring released. Find how high the weight rises before it leaves the spring and its velocity at that instant.
Prove that the middle points of a system of parallel chords of the curve \[ ax^2+2hxy+by^2=1 \] lie on a straight line through the origin. \par Show that the chord of this curve which has \((X, Y)\) for its middle point is \[ axX + h(xY+yX)+byY = aX^2+2hXY+bY^2. \]
Define a unit magnetic pole. How is a ``line of magnetic force'' defined by means of the unit pole? \par Two bar magnets, each of length 50 cms. and pole strength 50 units, are laid centrally across one another at right angles. Find the couple in dyne-centimetres on each magnet due to the other.
A segment of a circle is to have a given area, and the length of the chord of the segment together with \(n\) times the length of the arc is to be a minimum. Prove that if \(n>1\) the segment must be greater than a semicircle, and that the angle in the segment must have its secant equal to \(n\). \par What is the solution if \(n<1\)?