Problems

S=0, T=0, L=0 and M=0 are the equations of a conic, a tangent to the conic, a chord and a chord passing through the point of contact of T. Interpret the equations \[ S-LT=0, \quad S-MT=0, \quad S-T^2=0. \] Find the equation of the circle of curvature at the point \((x', y')\) on the conic \[ ax^2+by^2+c=0. \]

ABC is a triangle and the perpendiculars \(p,q,r\) from A, B, C to a variable straight line are such that \(p^2=\lambda qr\) where \(\lambda\) is a constant. Prove that the line envelops a conic which touches AB at B and AC at C.

Show that \(fyz+gzx+hxy=0\) is the equation in homogeneous coordinates of a conic circumscribing the triangle of reference. Find the equation of the chord joining the points \((x',y',z'), (x'',y'',z'')\) and deduce or otherwise find the equation of the tangent at \((x',y',z')\) in the form \[ fx/x'^2 + gy/y'^2 + hz/z'^2 = 0. \] Obtain the tangential equation of the curve.

Prove that, with a proper choice of a triangle of reference, the equation of a conic through four fixed points is of the form \(ax^2+by^2+cz^2=0\), where \(a,b,c\) are connected by a relation \(af^2+bg^2+ch^2=0\). \par Prove that the polars of a given point with regard to such a system of conics all pass through another point, and that if the first point lies on a given line the second lies on a conic which circumscribes the common self polar triangle of the system of conics, and find the equation of the conic.

By proving that the Simson Line of a point on the circumcircle of a triangle bisects the join of the point to the orthocentre, or otherwise, shew that the orthocentres of the four triangles formed by four straight lines are collinear, and that the four circumcircles have a common point. \par Apply this result to the case of four tangents to a parabola.

Shew that the inverse of a circle with respect to a point not necessarily in its plane is either a circle or a straight line. \par If \(P_1\) and \(P_2\) are a pair of points inverse with respect to the circle S, shew by inversion with respect to a coplanar point O not on S that they become \(P_1'\) and \(P_2'\), a pair of points inverse with respect to the inverse of S. \par State the result in the cases (i) when O does lie on S, and (ii) when O does not lie in the plane of S.

Prove that a circle through the vertex of a parabola cuts the curve again in three points at which the normals to the parabola are concurrent. \par Taking the co-ordinates of this point of concurrence as \((h,k)\) with the equation of the parabola as \(y^2=4ax\), find the co-ordinates of the centre of the circle.

Shew that the locus of the poles of a fixed straight line with respect to conics through four fixed points is also a conic. \par Apply this result to the case of a system of coaxal circles, and shew that one point of the resulting conic lies on the radical axis where it is cut by the circle orthogonal to the system, and passing through the intersection of the radical axis with the given line.

Taking the equation of a straight line as \(lx+my=1\), shew that the tangential equation \(Hlm+Ul+Vm=0\) represents a parabola. \par Find the co-ordinates of the focus and the equation of the directrix.

A conic, inscribed in a triangle ABC, touches BC, CA, AB, at A', B', C', respectively. Shew that if any other conic through A', B', C' cuts the sides of the triangle again in A'', B'', C'', then AA'', BB'', CC'' are concurrent.