A'A is the major axis of an ellipse of centre O and foci S', S. The tangent at a point P of the ellipse meets the major axis in T, the normal at P meets the major axis in G, and the perpendicular from P (the ordinate through P) meets it in N. Prove that \[ ON.OT=OA^2, \quad OG.OT=OS^2. \] Deduce that the eccentricity of the ellipse is \(\sqrt{(OG/ON)}\), and that the length of the minor semi-axis is \(\sqrt{(NG.OT)}\).
The tangents to the ellipse \(\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) at the points \(L_1, M_1\) meet at \(P_1\), and the tangents at \(L_2, M_2\) meet at \(P_2\). Prove that the six points \(L_1, M_1, P_1, L_2, M_2, P_2\) lie on a conic. Prove also that, if this conic is a rectangular hyperbola, then the points \(P_1, P_2\) are conjugate with respect to the director circle of the ellipse.
The normal to the parabola \(y^2=4ax\) at the point \(P(ap^2, 2ap)\) meets the curve again in \(N(an^2, 2an)\), and the line joining \(N\) to the focus \((a,0)\) meets the curve again in \(Q(aq^2, 2aq)\). Prove that \[ q = p/(p^2+2). \] Deduce that there are two positions of \(P\), say \(P_1\) and \(P_2\), which give rise in this way to the same point \(Q\), and prove that all the straight lines joining corresponding points \(P_1, P_2\) meet the axis of the parabola in a fixed point.
S is a fixed point outside a given circle of centre S'. An arbitrary point U is taken on the circle, and the perpendicular bisector of SU meets the line through S' and U in a point P. Prove that the locus of P is a hyperbola. Identify the two parts of the circle on which U must lie in order to give rise to the two branches of the hyperbola, and determine the four positions of U on the circle for which the corresponding point is at an end of a latus rectum.
Prove that the equation of the parabola which touches the rectangular hyperbola \(xy=c^2\) at each of the points U(\(cu, c/u\)), V(\(cv, c/v\)) is \[ x^2 - 2uvxy + u^2v^2y^2 - 2c(u+v)x - 2cuv(u+v)y + c^2(u^2+6uv+v^2)=0. \] The axis of this parabola cuts the principal axes of the hyperbola (\(x\pm y=0\)) in P and Q, and also cuts the line UV in R. Prove that R is the middle point of PQ and also that R is the foot of the perpendicular from the origin to UV.
Obtain the equation of a conic inscribed in the triangle of reference XYZ of general homogeneous coordinates in the form \[ x^2+y^2+z^2-2yz-2zx-2xy=0. \] The line joining X to the point of contact on YZ cuts the conic again in U; points V, W are defined similarly by cyclic interchange of the letters X, Y, Z. Prove that the conic UVWYZ touches the given conic at U.
Lines \(\alpha, \beta, \gamma\) are drawn through the respective vertices \(A, B, C\) of a triangle \(ABC\). Establish a necessary and sufficient condition for the concurrence of \(\alpha, \beta, \gamma\), in terms of the sines of the angles that these lines make with the sides of the triangle at its vertices. Points \(D, E, F\) are taken arbitrarily on the perpendiculars from a general point \(P\) to the sides \(BC, CA, AB\) respectively. Prove that the perpendiculars from \(A, B, C\) to \(EF, FD, DE\) respectively are concurrent.
Equilateral triangles \(BCD, CAE, ABF\) are constructed on the sides of a triangle \(ABC\) and external to this triangle. Prove that (i) the lines \(AD, BE, CF\) are concurrent; (ii) the circumcentres of the three given equilateral triangles are the vertices of another equilateral triangle.
\(A_1A_2A_3A_4\) is a tetrahedron and \(O\) is a point in general position. On each edge \(A_rA_s\) the point \(P_{rs}\) is the intersection of that edge with the plane through \(O\) and the opposite edge. Prove that the lines \(P_{13}P_{23}, P_{14}P_{24}\) and \(A_1A_2\) are concurrent at, say, \(Q_{12}\). If the corresponding point \(Q_{rs}\) is taken on each edge, prove that \(Q_{12}, Q_{13}, Q_{23}\) are collinear and that the six points \(Q_{rs}\) are coplanar.
A triangle \(ABC\) is inscribed in a circle \(\Sigma\) and circumscribed to a parabola \(\Gamma\). Prove that the focus \(S\) of \(\Gamma\) lies on \(\Sigma\). Another parabola \(\Gamma'\) with focus \(S'\) is inscribed in the triangle \(ABC\) and has its axis perpendicular to that of \(\Gamma\). Prove that \(SS'\) is a diameter of \(\Sigma\).