Prove that if CP, CD are conjugate semi-diameters of an ellipse whose foci are S and S', the rectangle SP.S'P is equal to the square on CD. Prove also that if SP and CD intersect in E, then \(CP^2-SE^2\) is equal to the square on the semi-minor axis.
Solve the equations \[ x(y+a)-ay = y(z+a)-az = z(x+a)-ax \] \[ 3(x+y+z)=10a. \] Resolve into partial fractions \[ \frac{1}{(1-2x)(1-8x^3)}. \]
Assuming that the series \[ 1+6x+12x^2+kx^3+120x^4+408x^5+\dots \] is a recurring series, determine the value which k must have and find the general term of the series.
Shew that the series \[ \frac{1}{1^p}+\frac{1}{2^p}+\frac{1}{3^p}+\dots \] is convergent only when \(p>1\). Discuss the convergency of the series \[ \frac{1^p}{2^q}+\frac{2^p}{3^q}+\frac{3^p}{4^q}+\dots \] where \(p\) and \(q\) are positive numbers.
Prove that any number of coplanar forces not in equilibrium can be reduced to a single force or a couple. If the forces \(P_1, P_2, P_3\), acting at points whose coordinates are \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) in directions making angles \(\theta_1, \theta_2, \theta_3\) with the axes of \(x\), reduce to a single force, find the equation of its line of action.
Explain the meaning of limiting friction and total resistance, and find the least force which will just pull a heavy body up an inclined plane. Shew that the greatest inclination to the horizon at which a uniform rod can rest in a rough sphere of radius \(a\), and angle of friction \(\lambda\), is \(\tan^{-1}\frac{a^2\sin\lambda\cos\lambda}{c^2-a^2\sin^2\lambda}\), where \(c\) is the distance of the rod from the centre of the sphere.
Five equal uniform rods AB, BC, CD, DE, EA are hinged together and the framework is supported with AB and BC in a horizontal line resting on two smooth pegs, and DE also horizontal. Shew that the distance between the pegs is \(1\frac{3}{5}\) times the length of a rod.
Prove that if the ends of each of two diagonals of a complete quadrilateral are conjugate points with respect to a given conic, the ends of the third diagonal will also be conjugate points.
Given two tangents to a conic with their points of contact and one other point of the conic, give a construction for the centre of the conic.
Prove that the locus of points whose tangents to the two conics \[ S = ax^2+by^2+cz^2=0, \quad S' = a'x^2+b'y^2+c'z^2=0, \] form a harmonic pencil, is \[ F = \Sigma aa'(bc'+b'c)x^2 = 0. \] Deduce that, for non-degenerate S, S', the invariant condition that F should degenerate into two straight lines is \(\Delta\Delta' = \Theta\Theta'\).