Obtain a geometrical construction for dividing a line into two parts so that the rectangle contained by the whole line and one part may be equal to the square on the other part. Shew how this mode of division may be used to construct a regular pentagon. Obtain a construction for dividing a line so that the square on one part may be \(n\) times the rectangle contained by the whole line and the other part, where \(n\) is any integer.
If \(P\) is a point in the plane of the triangle \(ABC\) and \(\alpha.PA^2 + \beta.PB^2 + \gamma.PC^2 = \delta\), where \(\alpha, \beta, \gamma, \delta\) are constants, prove that the locus of \(P\) is a circle. Hence, or otherwise, find the position of the point \(P\) when \(PB^2+PC^2-PA^2\) is a minimum; and shew that \(CA.PB^2+AB.PC^2-BC.PA^2\) is a minimum at the centre of one of the escribed circles of the triangle \(ABC\). State the minimum value in each case.
Prove Pascal's theorem, and shew by means of it how to construct any number of points on a conic
Shew how to find the focus, directrix and eccentricity of the section of a circular cone by any plane. If a circular cone of vertical angle \(2\alpha\) is cut by a plane so that the greatest and least distances of the curve of section from the vertex are \(r, r'\), shew that the minor semi-axis of the section is \((rr')^{1/2}\sin\alpha\). Hence shew that the envelope of planes which cut a given circular cone in sections having a minor axis of given length is a hyperboloid of revolution.
Shew how to draw a line through a given point to meet two given non-intersecting lines. If \(A, B, C\) are three non-intersecting lines shew that there is an infinity of lines \(L_r\) meeting all three. If the plane of \(L_r\) and \(A\) meets a fourth line \(D\) in \(P_r\), and the plane of \(L_r\) and \(B\) meets \(D\) in \(Q_r\), shew that there is in general a one-one correspondence between \(P_r\) and \(Q_r\) and hence that \(D\) meets either two or all of the lines \(L_r\).
Obtain the equation of the pair of tangents from a point \((x_1, y_1)\) to the ellipse \(\displaystyle\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). A point \(P\) moves so that the part of a fixed tangent to the ellipse intercepted between the tangents from \(P\) to the ellipse subtends a right angle at the centre. Shew that the locus of \(P\) is a straight line touching the ellipse \(\displaystyle\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{a^2+b^2}{a^2-b^2}\).
Shew that there are four normals from a point \((h,k)\) to the ellipse \(\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\); that the feet of these normals lie on a rectangular hyperbola passing through the centre of the ellipse; that two of these feet are always real points, one lying in the same quadrant with \(h,k\) and the other in the opposite quadrant, and that the other two, if real, are on the same side of the axis of \(y\) as the point \((h,k)\), and on the opposite side of the axis of \(x\). Shew that the normals at the four points at which the ellipse is met by the lines \[ \frac{mx}{a} + \frac{ny}{b} = 1, \quad \frac{x}{ma} + \frac{y}{nb} = -1 \] are concurrent in the point given by \[ \frac{ah}{a^2-b^2} = \frac{-n+1/n}{m/n+n/m}, \quad \frac{bk}{a^2-b^2} = \frac{m-1/m}{m/n+n/m}. \]
If four points on a rectangular hyperbola are such that the chord joining any two is perpendicular to the chord joining the other two shew that the same is true for the other pairs of chords joining the points. Shew also that the three pairs of chords joining any four points on a hyperbola cut off on either asymptote three segments which have a common middle point, and that all conics through the four points cut off segments from the asymptote with the same middle point.
Find the equation of the conic given by \[ x:y:1 = S_1(t):S_2(t):S_3(t), \] where \begin{align*} S_1(t) &= a_1t^2+b_1t+c_1, \\ S_2(t) &= a_2t^2+b_2t+c_2, \\ S_3(t) &= a_3t^2+b_3t+c_3. \end{align*} Shew that one of the asymptotes of the conic has the equation \[ xS_3(\alpha) - yS_1(\alpha) = (C\alpha^2-2B\alpha+A)/(\alpha-\beta)a_3, \] where \(\alpha, \beta\) are the roots of \(S_3(t)=0\), and \[ A=b_1c_2-b_2c_1, \quad B=c_1a_2-a_1c_2, \quad C=a_1b_2-a_2b_1, \] the other asymptote being obtained by interchanging \(\alpha\) and \(\beta\).
Shew that the tangential equation of all conics having the real points \((a,b)\) \((a',b')\) for foci is \[ (la+mb+1)(la'+mb'+1) - \lambda(l^2+m^2) = 0 \] and obtain the corresponding point equation. Shew that the hyperbolas of the system are given by negative values of \(\lambda\), and the ellipses by positive values.