Find the rhombus of maximum area and the rhombus of minimum area inscribed in the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
\(PSP'\), \(QSQ'\) are any two focal chords of a parabola. Shew that the common chord of the circles described on \(PP'\), \(QQ'\) as diameters passes through the vertex of the parabola.
A particle of mass \(m\) is suspended from a fixed point \(A\) by a light inextensible string of length \(a\) and is hanging at rest at \(B\) when it receives a horizontal impulse of magnitude \(m\left(\frac{4}{3}ag\right)^{\frac{1}{2}}\). Shew that in the subsequent motion the string becomes slack, the particle reaches a height \(27a/16\) above \(B\), and that when the string next becomes taut the particle is at \(B\).
A curve \(C\) on the earth's surface (assumed to be a sphere of radius \(a\)) cuts the meridians at a constant acute angle \(\alpha\). Prove that \[ \cos\lambda \cosh m\phi = 1, \] where \(m=\cot\alpha\), and \((\lambda, \phi)\) are the latitude and longitude, respectively, of a variable point \(P\) of \(C\), \(\phi\) being measured continuously along \(C\) from the point where \(C\) crosses the equator. Find (i) the length of the arc of \(C\) between the points \(P_1(\lambda_1, \phi_1)\) and \(P_2(\lambda_2, \phi_2)\), and (ii) the area swept out by the projection of \(OP\) on the equatorial plane, when \(P\) describes the arc \(P_1P_2\) of \(C\). Give each result in terms of the latitudes \(\lambda_1\) and \(\lambda_2\).
Prove that, if \(c_n\) is the coefficient of \((x+1)^n\) in the expansion of \[ \frac{e^{x^2+2x}}{(x^2+2x+2)^2} \] in a series of positive powers of \((x+1)\), then \(c_n=0\) if \(n\) is odd, while \[ c_{2k} = \frac{1}{2e} \sum_{m=0}^k \frac{(-1)^m(m+1)(m+2)}{(k-m)!} \quad (k=0, 1, 2, \dots). \]
Prove that an ellipse has two equal conjugate diameters. Shew further that the locus of the point of intersection of normals at the extremities of chords parallel to one of these is a straight line perpendicular to the other.
A wedge of mass \(M\) has two smooth plane faces inclined at an angle \(\alpha\), and is placed with one face in contact with a smooth horizontal table. A particle of mass \(m\) is placed centrally on the other face and allowed to slide down under gravity. Find the acceleration of the wedge and the force it exerts on the table.
Shew that a system of coplanar forces may be reduced (i) to a force acting at an assigned point \(P\) in the plane of the forces together with a couple \(G\), or (ii) to three forces acting along the sides of a given triangle \(ABC\) in the plane of the forces. What is the locus of \(P\) for a given value of \(G\)? If this locus passes through \(A\) and through the mid-point of \(BC\), and if the forces in (ii) are \(X\) in direction \(\vec{BC}\), \(5Z\) in direction \(\vec{CA}\), \(Z\) in direction \(\vec{AB}\), determine \(X\) in terms of \(Z\), it being given that the lengths of \(BC, CA, AB\) are in the ratio \(9:5:8\).
Prove that the polynomial \(X_n = \frac{d^n}{dx^n}(x^2-1)^n\) satisfies the equation \[ (1-x^2)\frac{d^2X_n}{dx^2} - 2x \frac{dX_n}{dx} + n(n+1)X_n = 0. \]
Shew that a circle drawn through the centre of a rectangular hyperbola and any two points will also pass through the intersection of lines drawn through each of these points parallel to the polar of the other.