Two parabolas have foci \(S_1\), \(S_2\), and the directrix of each passes through the focus of the other. Prove that each parabola touches the line which bisects \(S_1S_2\) at right angles, and that the two other common tangents are perpendicular to one another.
It is required to place forces in the sides of a given plane quadrilateral so that they shall have a resultant which is given in all respects. If forces \(P, Q, R, S\) be one solution, and \(P', Q', R', S'\) be a second solution, shew how to find the ratios that \(P-P'\), \(Q-Q'\), \(R-R'\), \(S-S'\) bear to each other.
The tangent and normal at a point \(P\) of a parabola whose focus is \(S\) meet the axis of the parabola in \(T\) and \(G\) respectively: prove that \(ST=SP=SG\). \par Prove also that, as \(P\) moves along the curve, \(GP^2 \propto GS\).
The prime factors of a number \(N\) are known, viz. \[ N = P_1^{a_1} P_2^{a_2} P_3^{a_3} \dots P_r^{a_r}, \] where \(P_1, P_2, P_3 \dots P_r\) are different prime numbers. Counting \(N\) itself and unity as divisors of \(N\), shew that the number of divisors of \(N\) is \[ (1+a_1)(1+a_2)(1+a_3)\dots(1+a_r), \] and find a formula for the sum of these divisors. Find also the number of ways in which \(N\) can be resolved into two factors prime to one another, and a formula for the sum of such factors. \par Prove that 60 is the smallest number which has 12 divisors; and generally that the smallest number which has \(k\) divisors is one of the numbers of the form \[ 2^{a_1} 3^{a_2} 5^{a_3} 7^{a_4} 11^{a_5} \dots, \] where \(a_1, a_2, a_3, \dots\) are numbers such that \(a_1 \ge a_2 \ge a_3 \ge a_4 \dots\), and \[ (1+a_1)(1+a_2)(1+a_3)\dots = k. \]
The middle point of a rod \(AB\) moves uniformly with given velocity in a circle, centre \(O\), and the end \(A\) moves in a straight line through \(O\) in the plane of the circle; shew graphically the velocity of \(B\).
Solve the equations \[ x (x-a) = yz, \quad y(y-b) = zx, \quad z(z-c) = xy. \]
A uniform solid hemisphere is placed with its curved surface in contact with a rough inclined plane. Shew that for equilibrium to be possible, the inclination of the plane to the horizontal must be less than \(\sin^{-1}(3/8)\). Shew, also, that there are two positions of equilibrium if the inclination of the plane lies between \(\tan^{-1}(3/8)\) and \(\sin^{-1}(3/8)\). Discuss the stability of the equilibrium, and shew that when one position only exists it is stable.
From the angular points \(A, B, C\) of an equilateral triangle, whose side is 3 inches, lines \(AP, BQ, CR\) are drawn, of lengths 1, 2, 4 inches respectively, perpendicular to the plane \(ABC\) and on the same side of it. Find where the plane \(PQR\) meets \(CB\) and \(BA\), and prove that the angle between the planes \(ABC\) and \(PQR\) is a little greater than \(45^\circ\).
Prove that the equation of a conic which touches the axis of \(x\) at the origin is of the form \[ ax^2+by^2+2hxy=2y; \] and that, near the origin, the curve lies on the positive or negative side of the axis of \(x\) according as \(a\) is positive or negative. \par Obtain the conditions that two conics, \[ ax^2+by^2+2hxy=2y, \quad a'x^2+b'y^2+2h'xy=2y, \] (i) should intersect in two real points, (ii) should have two real common tangents other than the axis of \(x\); and shew that the conditions are identical when \(a\) and \(a'\) are of the same sign.
Find the relation between the ``Watt'' and the ``Horse-power,'' given that 1 inch = 2.54 cms., and that 1 lb. = 453.6 grammes. \par An electric motor costs \pounds100 and runs for 1000 hours per annum: interest on the capital cost, depreciation and upkeep amount to 15\% of the cost per annum. If the average load be 10 H.P., if the average motor efficiency be 80\% and if electric energy costs 2d. per kilowatt-hour, find the total cost per horse-power-hour.