Prove that pairs of tangents from any point to conics touching four given straight lines form a pencil in involution. Show that the two parabolas touching the sides of a triangle and passing through a point \(P\) on its circumcircle cut orthogonally at \(P\).
If two triangles are both self polar with regard to a conic, prove that the six vertices lie on another conic. Show that the envelope of the axes of conics which touch the sides of a quadrilateral circumscribed about a circle is a parabola.
\(O\) is a fixed point; \(S, S'\) are two given conics. If \(A, A'\) are the poles with respect to \(S, S'\) of any line through \(O\), the line \(AA'\) envelopes a conic. Show that the envelope is a parabola if \(O\) lies on the line joining the centres of the conics.
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 \(OH\).