The straight lines \(AB\) and \(CD\) intersect in \(U\). \(AC\) and \(BD\) in \(V\); \(UV\) intersects \(AD\) and \(BC\) in \(F\) and \(G\) respectively; \(BF\) intersects \(AC\) in \(L\). Prove that \(LG, CF\) and \(AU\) meet in a point.
Prove that the polar reciprocal of a conic with regard to a focus is a circle. \par Find the number of conics which have a given focus and pass through three given points.
The tangents at points \(P\) and \(Q\) of a parabola meet at \(T\), and are of equal length. From a point \(U\) on \(TP\) another tangent is drawn, cutting \(TQ\) in \(V\). Prove that \(TV=PU\).
Any two conjugate diameters of an ellipse meet the tangent at one end of the major axis in \(Q\) and \(R\). Prove that \(QR\) subtends supplementary angles at the foci.
If \(O, D, E\) and \(F\) are the centres of the inscribed and escribed circles of a triangle, prove that \(O\) is the orthocentre of the triangle \(DEF\). Prove also that every conic passing through the points \(O, D, E, F\) is a rectangular hyperbola.
Tangents from \(P\) to a given circle meet the tangent at a given point \(A\) in \(Q\) and \(R\). If the perpendicular distance from \(P\) to the tangent at \(A\) is constant, prove that the rectangle \(QA.AR\) is constant.
Any tangent to the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) meets the ellipse \(\frac{x^2}{a}+\frac{y^2}{b}=a+b\) in \(P\) and \(Q\). Prove that the tangents at \(P\) and \(Q\) are perpendicular.
A conic has a focus at the centre of a given circle; its eccentricity, and the direction of its major axis are also given. Tangents are drawn to it at the points where it meets the circle. For all such conics, prove that these tangents envelope a conic whose major axis is a diameter of the given circle. \par What happens if the given eccentricity is unity?
The tangents from \(P\) to the conic \(ax^2+by^2=1\) are harmonic conjugates with respect to the tangents from \(P\) to the conic \(ax^2-cy^2=-1\). Prove that the locus of \(P\) is two parallel straight lines.
The equation of a conic in homogeneous coordinates is \[ ax^2+by^2+cz^2+2fyz+2gzx+2hxy=0. \] Find the condition that the pole of \(lx+my+nz=0\) should lie on \(l'x+m'y+n'z=0\).