Prove that the eight points of contact of the common tangents of two circles lie upon two straight lines, if the lengths of these common tangents are equal to the sum and difference of the radii of the two circles.
Prove that a system of conics passing through four fixed points \(A, B, C, D\) cuts
Prove that the polar reciprocal of a conic with regard to another conic is a conic, and determine the points which reciprocate into the asymptotes. Shew that four conics can be described passing through three given points \(A, B, C\) and having another given point \(S\) as focus, and that three of these conics are always hyperbolas. Prove that, if the fourth conic is an ellipse, the point \(C\) must lie within regions bounded by the two parabolas passing through \(A, B\) and having \(S\) as focus.
Prove that the tangents from any point to a sphere generate a circular cone. A variable small circle on a sphere is drawn to cut orthogonally a fixed small circle on the same sphere; prove that the plane of the variable circle passes through a fixed point.
Prove that the equation of any normal to the parabola \(y^2-4ax=0\) can be written in the form \[ y - mx + 2am + am^3 = 0. \] Find the locus of the foot of the perpendicular from the focus upon a variable normal of the parabola.
The tangents at the points \(P, Q\) of an ellipse, whose foci are \(S\) and \(H\), meet in \(T\). Prove that \[ SP \cdot SQ : ST^2 = HP \cdot HQ : HT^2, \] and that these two equal ratios are constant when \(T\) lies on a concentric, similar and similarly situated ellipse.
Find the equation of the rectangular hyperbola passing through the feet of the four normals which can be drawn from the point \((\alpha, \beta)\) to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} - 1 = 0\). Shew that the circle through the feet of three of these normals cuts the ellipse at the other end of the diameter through the foot of the fourth normal.
Find the equation of the system of conics confocal with the conic given by the equation \(6x^2-4xy+9y^2-1=0\), the axes of coordinates being rectangular; and determine the two conics of the system passing through the point \((2,1)\). Interpret the results.
Find the equation of the asymptotes of the conic given by the equation \(ax^2+by^2+cz^2=0\), the coordinates being areal.
Eliminate \(x\) and \(y\) from the equations \[ ax^2+by^2=1, \quad a'x^2+b'y^2=1, \quad lx+my=1, \] obtaining the result in a rational form.