Find the equation of the bisectors of the angles between the straight lines \[ Ax^2+2Hxy+By^2=0. \] If one of the bisectors of the angles between the tangents from a point \(P\) to an ellipse passes through a fixed point on the major axis, show that \(P\) lies either on the major axis or on a fixed circle whose centre is on the major axis.
Find the equation of the circle circumscribing the triangle formed by the lines \[ x=0, \quad y=0, \quad \frac{x}{a}+\frac{y}{b}=1. \] Prove that the tangents to this circle at the angular points meet the opposite sides on the line \[ a^3x+b^3y+a^2b^2=0. \] % Note: OCR had a^2 x + b^2 y + a^2 b^2 = 0. Corrected to a^3, b^3 as is more likely from symmetry.
If the lines \(lx+my=1, l'x+m'y=1\) are conjugate (i.e. each passes through the pole of the other) with respect to the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), show that \(a^2ll'+b^2mm'=1\). If pairs of conjugate lines are drawn through the ends of the major axis of the ellipse, show that the locus of the intersection of the pairs is the ellipse \(\frac{x^2}{a^2}+\frac{2y^2}{b^2}=1\).
Show that \(x=at^2+2bt, y=a't^2+2b't\) represents a parabola, \(t\) being a variable parameter. Find the condition that the line \(y-y_0=m(x-x_0)\) may touch the parabola and deduce that the directrix is \(ax+a'y+b^2+b'^2=0\).
Interpret the equation \(S=L^2\), where \(S\) is of the second degree in \(x,y\) and \(L\) is of the first degree in \(x,y\). A variable conic touches \(ax^2+by^2=1\) at points on the line \(lx+my=1\). Show that the locus of the points of contact of tangents to the conic from the origin is \(ax^2+by^2=lx+my\).
Using areal (or trilinear) coordinates, find the coordinates of the centre of a conic circumscribing the triangle of reference. Two conics circumscribe a triangle and touch one another at one of the angular points. Prove that their centres, their point of contact, and the middle points of the sides lie upon a conic.
The bisector of the angle \(BAC\) of a triangle \(ABC\) cuts the circumcircle of the triangle in \(D\). Prove that the straight line joining the feet of the perpendiculars from \(D\) on the sides \(AB\) and \(AC\) bisects the side \(BC\).
Prove that, if four collinear points \(A, B, C, D\) form a harmonic range, and \(O\) is the middle point of \(AC\), then \(OC\) is the mean proportional between \(OB\) and \(OD\). \(P\) is a point on the side \(BC\) of a triangle \(ABC\), and \(Q\) is a point on \(AP\); if \(BQ\) cuts \(AC\) at \(R\), \(CQ\) cuts \(AB\) in \(S\) and \(RS\) cuts \(BC\) in \(T\), prove that the points \(B, P, C, T\) form a harmonic range.
Prove that the tangents to a parabola at the extremities of a focal chord intersect in the directrix. A chord \(PP'\) of a parabola cuts the axis in \(Q\) and the tangents at \(P\) and \(P'\) intersect in \(T\); prove that \(QT\) is bisected by the tangent at the vertex.
Shew that the feet of the perpendiculars from the foci on the tangent at any point of an ellipse lie on a fixed circle. Shew also that the ordinate of the point bisects the angle between the lines joining the foot of the ordinate and the feet of the perpendiculars.