Prove that \(v^2/\rho\) is the acceleration inwards along the normal, when a particle describes a plane curve with velocity \(v\). A smooth parabolic tube has its axis vertical and vertex upwards. A particle inside the tube is projected from the vertex with a velocity \(\sqrt{2g(a+h)}\). When the particle is at a depth \(z\) below the directrix prove that the pressure on the tube is \(mgh\sqrt{az^{-3}}\), where \(m\) is the mass of the particle and \(4a\) is the latus rectum of the parabola.
The sides of a plane polygon \(A_1A_2A_3\dots A_n\) are cut by a straight line in the points \(B_1, B_2, B_3, \dots B_n\) where \(B_r\) is the point of intersection of the line with the side \(A_rA_{r+1}\): prove that \[ A_1B_1 . A_2B_2 \dots A_nB_n = A_2B_1 . A_3B_2 \dots A_1B_n. \] Shew that the same equality holds if the polygon is not a plane one and the sides are intersected by a plane.
Shew that any transversal cuts a plane pencil of four fixed lines in a range of constant anharmonic ratio. The points \(A, B, C\) on a line \(OA\) and the points \(A', B', C'\) on a line \(OA'\) are such that \(AA', BB', CC'\) meet in a point: shew that if the line \(OA'\) be turned about \(O\) through any angle in the plane, the lines \(AA', BB', CC'\) in their new positions meet in a point and the locus of this point for different angles is a circle.
Four points lie on a circle: shew that the six perpendiculars, each drawn from the middle point of a chord joining two points to the chord joining the other two, pass through the same point the line joining which to the centre of the circle is bisected by the centroid of the four points. Prove that the point of intersection of the perpendiculars is the centre of the rectangular hyperbola through the four points and give the limiting form of the theorem when three of the four points coincide so that the circle is the circle of curvature at a point of the rectangular hyperbola.
Prove that the circle circumscribing the triangle formed by three tangents to a parabola passes through the focus. The tangent at a point on an ellipse meets the axes in \(T\) and \(t\) and the normal at the point meets them in \(G\) and \(g\): prove that the parabola which touches the tangent, normal and the axes of the ellipse has its focus at the intersection of \(Tg\) and \(Gt\) and touches the normal in the centre of curvature of the ellipse.
A line moves in a plane so that the product of the lengths of the perpendiculars on the line from two fixed points \(S\) and \(H\) in the plane is constant: prove that the feet of the perpendiculars lie on a circle and that the envelope of the line is the locus of a point \(P\) such that \(SP+PH\) or \(SP \sim PH\) is constant. Prove that the tangents to two circles at their common points touch a conic with its foci at the centres of the circles.
Interpret the equations:
Shew that any triangle inscribed in the parabola \(y^2=ax\) so that its centroid is at the fixed point \(x=2c, y=0\) is self-conjugate with regard to the parabola \(y^2=a(3c-2x)\) and circumscribed to the parabola \(y^2=4a(x-3c)\).
Shew that the angle between the central radius and the normal in an ellipse of semi-axes \(a\) and \(b\) has its greatest value \(2\tan^{-1}\frac{a-b}{a+b}\) when the inclination of the normal to the major axis is \(\tan^{-1}\sqrt[4]{\frac{a}{b}}\): shew also that the normal has its greatest distance \((a-b)\) from the centre when the inclination is \(\tan^{-1}\frac{\sqrt{a}}{\sqrt{b}}\).
Prove that the diameters of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), which lie along the lines \(y=mx, y=nx\), are conjugate if \(amn+b^2=0\). A pair of conjugate diameters of an ellipse have lengths \(2c_1, 2c_2\) and the parallel diameters of a second ellipse have lengths \(2d_1, 2d_2\), the axes of the two ellipses not necessarily being parallel: prove that \(\frac{c_1^2}{d_1^2}+\frac{c_2^2}{d_2^2}\) is constant, whatever pair of conjugate diameters be taken in the first ellipse.