Two parabolas have a common focus, and their axes lie in opposite directions along the same line. The polars of any point \(P\) with respect to the two parabolas intersect in \(Q\). Show that \(P,Q\) are equidistant from the common chord of the parabolas.
Find the equation of the pair of tangents from a given point to the conic \[ ax^2+by^2=1. \] A pair of tangents to any confocal of \(\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) pass each through fixed points on the axis of \(y\). Show that the locus of their intersection is a circle.
Find the tangential equation of the conic \[ ax^2+by^2+cz^2+2fyz+2gzx+2hxy=0, \] and determine the tangents common to \begin{align*} x^2+y^2+4zx-2xy &= 0, \\ 16x^2-3y^2+5z^2-2yz &= 0, \end{align*} and the tangential equation of one of the points of contact with the first of these conics.
Interpret the equation \(S=kL^2\), where \(S=0\) is a conic and \(L=0\) a line. A variable circle has double contact with the ellipse \(ax^2+by^2+c=0\). Prove that the chord of contact is parallel to one of the axes and that the envelope of the polar of a fixed point with respect to the circle consists of two parabolas.
Points \(P,Q,R,S\) are taken on the sides \(AB,BC,CD,DA\) of a square \(ABCD\) such that the figure \(PQRS\) is a rectangle. Prove that either (i) \(PQRS\) is a square, or (ii) \(PQRS\) has its sides parallel to \(AC\) and \(BD\).
Points \(X,Y,Z\) are taken on the sides \(BC,CA,AB\) of a triangle \(ABC\), and the circumcircle of the triangle \(XYZ\) cuts the sides \(BC,CA,AB\) again in \(X',Y',Z'\). Find the relations between the angles of the triangles \(XYZ\) and \(X'Y'Z'\). Prove that, if the triangles \(ZXY\) and \(Y'Z'X'\) are congruent they are similar to the triangle \(ABC\).
Any point \(O\) is taken on the circumcircle of a triangle \(ABC\); \(X,Y,Z\) are the projections of \(O\) on \(BC,CA,AB\); and \(OX\) meets the circle again in \(P\). Prove that \(X,Y,Z\) lie on a line that is parallel to \(AP\). Prove also that, if \(H\) is the orthocentre of the triangle \(ABC\), the line \(XYZ\) bisects \(OH\).
Prove that, if opposite edges of a tetrahedron are equal, the line joining the mid-points of any pair of opposite edges is perpendicular to them. Prove that, provided \(ABC\) is an acute-angled triangle, it is possible to construct a point \(D\) such that the tetrahedron \(ABCD\) has its opposite edges equal.
Prove that, if the circle drawn with centre \(O\) and passing through the focus \(S\) of a parabola cuts the directrix in \(M\) and \(N\), the lines through \(O\) that are perpendicular to \(SM\) and \(SN\) are tangents to the parabola. Hence, or otherwise, prove that, if \(P\) and \(Q\) are the points of contact of the tangents drawn to the parabola from \(O\), the triangles \(SOP\) and \(SQO\) are similar.
Prove that, if a parallelogram is circumscribed to an ellipse, its diagonals are conjugate diameters of the ellipse. Prove that, if a pair of conjugate diameters of an ellipse meet the director circle in \(P\) and \(Q\), the line \(PQ\) touches the ellipse.