Problems

\(A\) and \(B\) are two fixed points on a fixed circle. \(PQ\) and \(P'Q'\) are a variable pair of chords parallel to and equidistant from the chord \(AB\). Shew that the lines \(PP', PQ', QP'\) and \(QQ'\) touch a fixed parabola, which is also touched by the tangents to the circle at \(A\) and \(B\). What is the projective generalisation of this theorem?

Explain what is meant by an involution on a conic and shew that the joins of pairs of points of an involution pass through a fixed point. \(A_1, A_2, \dots, A_6\) are given points on a given conic. Shew that the six points \(A_5, A_6, (15, 26), (16, 25), (35, 46)\) and \((36, 45)\) lie on a conic, where \((15, 26)\) for instance denotes the point of intersection of \(A_1A_5\) and \(A_2A_6\). Prove also that the two remaining intersections of the two conics are the points of contact of tangents to the given conic from the point \((12, 34)\).

Two circles, with centres \(A\) and \(B\) and radii \(a\) and \(b\), lie in different planes which meet in a line \(l\). Shew that the two circles will lie on the same sphere provided that \(AB\) is perpendicular to \(l\) and that \[ AP^2 - BP^2 = a^2 - b^2, \] where \(P\) is any point on \(l\). Shew that these conditions are always satisfied if the two circles have two common points.

\(PQ\) 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 \[ 2T(Tb-Wa)^2 = P^2Wab. \]