Prove that the polar reciprocal of a circle \(C\) with respect to another circle \(K\) is a conic \(C'\) whose focus is the centre of \(K\) and whose directrix is the polar with respect to \(K\) of the centre of \(C\).
Determine the length of the perpendicular let fall from any point \((h,k)\) on the line \(ax+by+c=0\). Prove that the product of the perpendiculars from \((h,k)\) on the \(n\) straight lines represented by the equation \[ a_0x^n + a_1x^{n-1}y + a_2x^{n-2}y^2 + \dots + a_ny^n = 0 \] is \[ \frac{a_0h^n + a_1h^{n-1}k + a_2h^{n-2}k^2 + \dots + a_nk^n}{\sqrt{(\{a_0-a_2+a_4-\dots\}^2 + \{a_1-a_3+a_5-\dots\}^2)}}. \]
Find the equation of the director circle of the conic \[ ax^2+2hxy+by^2+2gx+2fy+c=0, \] the axes being rectangular.
Prove that through any point two conics confocal with \(x^2/a^2+y^2/b^2=1\) can be drawn and express the coordinates of the point in terms of the semi-axes of the confocals. Show that the locus of a point such that the tangents from it to the ellipse contain an angle \(2\alpha\) is given by the equation \[ a_1^2 \cos^2\alpha + a_2^2 \sin^2\alpha = a^2, \] where \(a_1, a_2\) are the primary semi-axes of the confocals through the point.
Find the equations of the tangent and normal at any point of the conic \[ l/r = 1+e\cos\theta. \] A system of conics have a common focus and directrix. Show that the normals at the points where a line through the focus in a given direction meets the conics envelope a parabola, having its vertex at the focus and touching the given line.
Prove that the general equation of a circle in areal coordinates is \[ (x+y+z)(t_1^2x+t_2^2y+t_3^2z) - (a^2yz+b^2zx+c^2xy) = 0, \] where \(a, b, c\) are the lengths of the sides of the triangle of reference \(ABC\) and \(t_1, t_2, t_3\) are the lengths of the tangents from \(A, B, C\) to the circle.
\(n\) equal, uniform, straight, smoothly jointed rods \(A_0A_1, A_1A_2, A_2A_3, \dots, A_{n-1}A_n\), are suspended from the end \(A_0\) while the end \(A_n\) can move on a fixed smooth vertical wire. The whole system is in equilibrium in a vertical plane with the rods inclined to the horizontal at angles \(\theta_1, \theta_2, \theta_3, \dots, \theta_n\) respectively. Prove that \[ (2n-1)\cot\theta_1 = (2n-3)\cot\theta_2 = (2n-5)\cot\theta_3 = \dots = \cot\theta_n = \frac{2R}{W}, \] where \(W\) is the weight of each rod and \(R\) the reaction at \(A_n\).
Define a couple and establish the principal properties of a couple. The figure represents the horizontal section through a door \(OA\) which can move between the two extreme perpendicular positions \(OX, OY\) about hinges in the vertical straight line through \(O\). \(AY\) is a light extensible string which obeys Hooke's Law. A spring exerts a restoring couple proportional to \(\theta\) and the hinges produce a constant frictional couple \(F\) when the door is in motion. It is found that the work done in slowly opening the door from \(OX\) to \(OY\) is equal to the work done in slowly closing the door again, whereas the work done in slowly half opening the door is \(\frac{1}{n}\)th of the work done in slowly closing the door from this position. Shew that if the string \(AY\) were cut, then the door would remain ajar provided \(\theta\) were not greater than \(\frac{3}{8}\frac{n+1}{n-1}\pi\). It may be assumed that \(n > 7\) and that the unstretched length of the string lies between \(2OA\sin\frac{\pi}{8}\) and \(2OA\sin\frac{\pi}{4}\).
Prove that a uniform solid elliptic cylinder can be in equilibrium on a rough inclined plane with its generators horizontal provided the eccentricity \(e\) of the normal cross-section of the cylinder is not less than \(\tan\alpha (2\operatorname{cosec}\alpha - 2)^{1/2}\), where \(\alpha\) is the inclination of the plane to the horizontal. If this condition is satisfied how many distinct positions of equilibrium exist? Determine the stability, or otherwise, of any possible positions of equilibrium in the case \[ \tan\alpha = \frac{\sqrt{2}}{8}, \quad e = \frac{\sqrt{2}}{2}. \]
Define the bending moment and shearing stress at a point of a beam. Draw the bending moment and shearing stress diagrams for a uniform horizontal beam \(AC\) of weight \(W\) which is freely hinged at the end \(A\) and rests on a support at \(B\) where \(AB=2BC\), and which supports a weight \(W\) at \(C\).