Shew that an infinite number of triangles may be inscribed in the parabola \(y^2=4ax\) so as to be self-conjugate with respect to the parabola \[ y^2+8ax+2py+q=0 \] for all values of \(p\) and \(q\). Shew that when \(q=\dfrac{p^2}{3}\) the two parabolas touch at the point \(\left(\dfrac{p^2}{36a}, -\dfrac{p}{3}\right)\); and further that when \(p\) varies the vertices of the parabolas \[ y^2+8ax+2py+\frac{p^2}{3}=0 \] describe the parabola \(y^2=12ax\).
(a) Shew that of the conics through four general points of a plane, two are parabolas, and one a rectangular hyperbola. \item[] (b) A conic touching an asymptote of a hyperbola at a point \(O\) meets the hyperbola in \(P, Q, R\) and \(S\). Prove that all conics through these four points meet that asymptote in two points equidistant from \(O\).
Find the equation referred to its principal axes of the conic \[ 11x^2+96xy+39y^2-74x+18y-71=0, \] and determine the eccentricity.
Find the angle between the lines \[ ax^2+2hxy+by^2=0, \] and the condition that two of the lines \[ ax^3+3bx^2y+3cxy^2+dy^3=0 \] should be at right angles. Find the relation between \(n\) and \(p\) in order that the locus may be a rectangular hyperbola when a point moves so that the sum of the squares of its distances from \(n\) fixed points is equal to the sum of the squares of its distances from \(p\) fixed lines.
Find the eight points of contact of common tangents to the conics whose equations in homogeneous coordinates are \begin{align*} x^2+y^2+z^2&=0, \\ ax^2+by^2+cz^2&=0 \end{align*} and shew that they lie on a conic.
A variable tangent to a conic meets the tangents at two fixed points \(A\) and \(B\) in \(P\) and \(Q\) respectively. Shew that the point of intersection of \(AQ\) and \(BP\) describes a conic having double contact with the given conic at \(A\) and \(B\). What does this become when \(AB\) is the line at infinity? Give an independent proof for this case.
A solid of uniform density consists of a solid cone of height \(h\) to the base of which is attached symmetrically a solid hemisphere of radius \(a\) equal to the radius of the base of the cone. Shew that the body can rest in stable equilibrium with its spherical surface in contact with a rough plane inclined at an angle \(\alpha\) to the horizontal, provided \(\dfrac{3a^2-h^2}{4(h+2a)} > a\sin\alpha\). Shew, further, that in such a position the plane of the base of the cone is inclined at an angle \(\beta\) to the horizontal, where \(\sin\beta = \dfrac{4a(h+2a)}{3a^2-h^2}\sin\alpha\).
A heavy uniform chain of line density \(w\) hangs over a rough circular cylinder of radius \(a\) having its axis horizontal and perpendicular to the plane of the chain. Obtain in the case of limiting equilibrium the equation involving the value of the tension \(T\) at a point of the chain in contact with the cylinder, in the form \[ \frac{dT}{d\theta} - \mu T = wa\sec\lambda\cos(\theta-\lambda), \] where \(\mu=\tan\lambda\) is the coefficient of friction and \(\theta\) is measured from the point at which the shorter end of the chain comes into contact with the cylinder. Hence prove that if the shorter free length is just zero, the greatest length possible on the other side is \(a\sin 2\lambda(1+e^{\mu\pi})\).
The ends of a heavy uniform rod of length \(a\) are constrained by rings to move on a rough circular wire of radius \(a\) fixed in a vertical plane. If \(\mu\), the coefficient of friction, is less than \(\sqrt{3}\), shew that the greatest inclination, \(\theta\), of the rod to the horizontal when in equilibrium is \(\tan^{-1}\dfrac{4\mu}{3-\mu^2}\).
A regular pentagon \(ABCDE\) consists of heavy uniform rods each of weight \(W\) freely jointed at their ends. The whole rests in a vertical plane with \(AB\) in contact with a smooth horizontal table, rigidity being maintained by two light struts \(AD, BD\). Calculate the thrust in these struts.