\(PQ\) is a focal chord of a parabola and the normals at \(P\) and \(Q\) meet the parabola again at \(P'\) and \(Q'\). Shew that \(PQ\) and \(P'Q'\) are parallel and that the ratio of their lengths is \(1:3\).
Shew that the equation in rectangular cartesian coordinates of any conic through the vertices of the triangle formed by the lines \[ l_1x+m_1y+n_1=0, \quad l_2x+m_2y+n_2=0, \quad \text{and} \quad l_3x+m_3y+n_3=0 \] is of the form \[ \frac{a}{l_1x+m_1y+n_1} + \frac{b}{l_2x+m_2y+n_2} + \frac{c}{l_3x+m_3y+n_3} = 0, \] and that this conic is a circle if \[ a:b:c = (l_1^2+m_1^2)(l_2m_3-l_3m_2) : (l_2^2+m_2^2)(l_3m_1-l_1m_3) : (l_3^2+m_3^2)(l_1m_2-l_2m_1). \] Find the condition that the feet of perpendiculars from the origin on to the sides of the triangle shall be collinear.
Shew that the equation of the chord of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) which is perpendicularly bisected by the line \(lx+my=1\) is \(c^2\left(\frac{x}{l}-\frac{y}{m}\right) = \frac{a^2}{l^2} + \frac{b^2}{m^2}\), where \(c^2=a^2-b^2\). If the line \(lx+my=1\) touches the parabola \[ (x-y)^2 - 2\kappa(x+y)+\kappa^2=0, \] shew that the chord touches the parabola \[ c^4(x+y)^2 + 4\kappa c^2(b^2x-a^2y) = 4a^2b^2\kappa^2. \]
Find the equation of the conic \(\Sigma\) which passes through \((x_1, y_1)\) and has double contact with the conic \[ S \equiv ax^2+2hxy+by^2+2gx+2fy+c=0 \] along the line \(\lambda x + \mu y = 0\). Shew that as \(\lambda\) varies the locus of the centre of \(\Sigma\) is the conic whose equation is \[ (S_1-gx-fy-c)^2 = S_{11}(S-gx-fy-c), \] where \begin{align*} S_1 &= ax_1x+h(x_1y+xy_1)+by_1y+g(x_1+x)+f(y_1+y)+c, \\ \text{and } S_{11} &= ax_1^2+2hx_1y_1+by_1^2+2gx_1+2fy_1+c. \end{align*}
Shew that by a suitable choice of the triangle of reference the equations of a pencil of conics through four distinct fixed points may be taken as \[ \lambda x^2 + \mu y^2 + \nu z^2 = 0, \] where \(\lambda, \mu\) and \(\nu\) vary subject to the relation \[ \lambda a^2 + \mu b^2 + \nu c^2 = 0. \] Find the equation of the tangent at \((x_1, y_1, z_1)\) to the conic of the pencil which passes through \((x_1, y_1, z_1)\), and shew that the locus of points of contact of tangents from \((x_1, y_1, z_1)\) to conics of the pencil is a curve whose equation is \[ a^2yz(y_1z-z_1y)+b^2zx(z_1x-x_1z)+c^2xy(x_1y-y_1x)=0. \] Shew that this curve passes through \((x_1, y_1, z_1)\), the four common points of the pencil of conics, and the vertices of the common self polar triangle.
A uniform elliptic lamina of axes \(2a, 2b\) rests, with its plane vertical, on two small smooth pegs, at a distance \(c\) apart, in the same horizontal line. Prove that, if \(a > c/\sqrt{2} > b\), there is a position of equilibrium in which the pegs are at ends of conjugate diameters.
A uniform rod \(AB\) of length \(2a\) is freely pivoted at the fixed end \(A\). A small smooth ring of weight \(P\) can slide on the rod, and is attached by a light string to a fixed point \(C\) vertically above \(A\). \(AC=b\). If in the position of equilibrium the rod and string are equally inclined to the vertical, prove that the tension \(T\) of the string is given by \[ 2T(Tb-Wa)^2 = P^2Wab. \]
Two weights \(P\) and \(Q\) are resting, one on each of two equally rough inclined planes, and are connected by a light smooth string passing over the horizontal line of intersection of the planes. The two portions of the string lie in lines of greatest slope of the planes. If \(Q\) is on the point of motion downwards, prove that the greatest weight that can be added to \(P\) without disturbing equilibrium is \[ \frac{P \sin 2\lambda \sin(\alpha+\beta)}{\sin(\alpha-\lambda)\sin(\beta-\lambda)}, \] where \(\alpha, \beta\) are the inclinations of the planes, and \(\lambda\) is the angle of friction.
A railway wagon of mass 8 tons, travelling at 8 feet per second, collides with a similar stationary wagon. Each of the four buffer springs in contact exerts a force of 2 tons when the buffers are fully extended, and requires an additional force of 1 ton for each inch of compression. The motion of a buffer in either direction is also resisted by a constant frictional force of \(\frac{1}{2}\) ton. Prove that each buffer is compressed 3 inches, and that the speeds of the wagons when contact ceases are \((4\pm 2\sqrt{3})\) feet per second.
A small smooth heavy ring is free to slide on a fixed parabolic wire whose axis is vertical and vertex downwards. The ring is projected from the vertex \(A\) with velocity \(\sqrt{2gh}\), and after passing the extremity \(B\) of the arc proceeds to describe an equal parabola freely. If \(c\) is the vertical height of \(B\) above \(A\), prove that the latus rectum is \(4(h-2c)\).