Problems

\(A,B,C\) are three points on a rectangular hyperbola. Prove that the orthocentre of the triangle \(ABC\) lies on the hyperbola at the other end of the diameter through the fourth point in which the circumcircle of \(ABC\) intersects the hyperbola. Shew also that the pedal triangle of \(ABC\) is self-conjugate with respect to the hyperbola.

State the connection between the foci of a conic and the circular points at infinity. \((x_1,y_1), (x_2,y_2)\) are the real foci of a conic which is reciprocated with respect to the circle \(x^2+y^2=a^2\). Shew that the equation of the reciprocal is \[ k(x^2+y^2) = (xx_1+yy_1-a^2)(xx_2+yy_2-a^2), \] where \(k\) is a constant. If the first conic passes through the origin, find an equation for \(k\).

\(t\) is the tangent to a given conic at a fixed point \(O\). \(P\) is a variable point such that the tangents from \(P\) to the conic intercept a constant length on \(t\). By taking the tangent and normal at \(O\) as axes, or otherwise, prove that the locus of \(P\) is a conic touching the given conic at the opposite end of the diameter through \(O\).

Prove that the family of conics passing through four general points in a plane is cut by any straight line in pairs of points in involution. \(P,Q\) are the points of contact of a common tangent to two given conics. \(R,S\) are the points in which this tangent is cut by a pair of common chords of the conics which do not intersect at a point of intersection of the conics. Prove that \(PRQS\) is a harmonic range.

If \(B'C', C'A', A'B'\) are respectively the polars of three non-collinear points \(A,B,C\) with respect to a conic, prove that \(AA', BB', CC'\) are concurrent. Examine the case when \(A'\) lies in \(BC\), \(B'\) in \(CA\), \(C'\) in \(AB\).

Establish conditions under which it shall be possible to obtain two distinct triangles having one side and one angle in common and the side opposite the given angle of given length. If \(x,y,z\) are respectively the distances between the circumcentres, the orthocentres, and the incentres, shew that \(2x=y+2z\).

(i) Express \(1-\cosh^2a-\cosh^2b-\cosh^2c+2\cosh a \cosh b \cosh c\) as the product of four sinh functions. (ii) If the inverse cosh is taken to be positive and if \[ \cosh^{-1}(x+y) + \cosh^{-1}(x-y) = \cosh^{-1}\lambda, \text{ where } x>y, \] express \(y^2\) in terms of \(\lambda\) and \(x\).

\(ABC\) is a triangle whose angle \(A\) is a right angle. Lines parallel to the opposite sides are drawn through \(B\) and \(C\), meeting the external bisector of the angle \(A\) in points \(L\) and \(P\) respectively. Show that \(BP, CL\) intersect on the perpendicular from \(A\) to the opposite side \(BC\).

\(A_1B_1C_1D_1\) and \(A_2B_2C_2D_2\) are two quadrangles such that the lines \(A_1B_1, C_1D_1, A_2B_2, C_2D_2\), meet in a point \(X\), and the lines \(B_1C_1, A_1D_1, B_2C_2, A_2D_2\), meet in a point \(Y\). A conic through \(A_1, B_1, C_1, D_1\) meets \(XY\) in two points \(P, Q\). Show, by means of involution properties or otherwise, that a conic can be drawn through the six points \(A_2, B_2, C_2, D_2, P, Q\). If \(A_1, C_1, B_2, D_2\) are collinear, show that \(A_2, C_2, B_1, D_1, X, Y\) lie on a conic.

Define conjugate points with respect to a conic, and show that the locus of points conjugate to a given point is a straight line. \(C\) is the pole of the line joining two points \(X, Y\) on a conic, and \(P\) is any other point of the conic. Show that \(PX, PY\) harmonically separate \(PC\) and the tangent at \(P\) to the conic. Interpret this result (a) when the tangent at \(P\) is the line at infinity, (b) when \(X, Y\) are points at infinity in two perpendicular directions.