Problems

(a) Give a geometrical construction for a circle through two given points which intercepts a given length on a given straight line. \item[] (b) \(A, O\) and \(B\) are three fixed points; two circles are drawn, one through \(A\) and \(O\), the other through \(O\) and \(B\), so as to cut at a constant angle. Find the locus of the other point of intersection.

Prove that the two tangents to an ellipse from an external point subtend equal angles at a focus. \(T\) is any point on the tangent to an ellipse at the extremity of its minor axis. Prove that the other tangent from \(T\) to the ellipse also touches the circle through \(T\) and the two foci.

Shew that two conics of a confocal system pass through an arbitrary point of the plane, and that one confocal touches an arbitrary line. Two straight lines meet on the line joining the foci, and two conics of the system are drawn touching the two lines in \(A\) and \(B\) respectively. Shew that the two lines which touch at \(A\) and \(B\) the other confocals through these points also meet on the line joining the foci.

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 \(\sin\beta = \dfrac{4a(h+2a)}{3a^2-h^2}\sin\alpha\).