Prove that the anharmonic ratio of the pencil formed by joining a variable point on a conic to four fixed points on the conic is constant. Two fixed points \(A,B\) are taken on a hyperbola and a variable point \(P\). A line \(AQR\) is drawn parallel to one asymptote meeting \(PB\) in \(Q\) and the parallel through \(P\) to the other asymptote in \(R\). The tangent at \(B\) and the parallel through \(B\) to the latter asymptote meet \(AR\) in \(T\) and \(S\) respectively. Prove that \(AT:AS=AQ:AR\).
Prove that the tangents from a point to an ellipse are equally inclined to the lines drawn from the point to the foci. Given a focus and two tangents to an ellipse, shew that the minor axis touches a parabola.
Prove that the points determined by the equations \[ ax^2+2hx+b=0, \quad a'x^2+2h'x+b'=0 \] will be harmonically conjugate if \(ab'+a'b-2hh'=0\). Shew that in general the anharmonic ratio \(\lambda\) of the range determined by these four points is given by \[ \left(\frac{1+\lambda}{1-\lambda}\right)^2 = \frac{(ab'+a'b-2hh')^2}{(h^2-ab)(h'^2-a'b')}. \]
Find the equations of the tangent and normal at the point \((at^2, 2at)\) of the parabola \(y^2=4ax\). The normals at \(P,Q\), the extremities of a focal chord of this parabola, meet the parabola again in \(P'Q'\). Prove that the envelope of the chord \(P'Q'\) is the parabola \(y^2=32a(9a-x)\).
Prove that the chords of intersection of a circle and an ellipse are equally inclined to the axes. A circle is drawn to touch the ellipse \(x^2/a^2+y^2/b^2=1\) at a point \(P\) and such that it intersects the ellipse in two points at the extremities of a diameter of the ellipse. Prove that the radius of the circle is \(\frac{a^2+b^2}{2ab}CD\), where \(CD\) is the diameter conjugate to \(CP\).
Find the condition that the lines \(y=mx, y=m'x\) should be parallel to conjugate diameters of the conic \(ax^2+2hxy+by^2+2gx+2fy+c=0\). If \(ax+hy+g=X\) and \(hx+by+f=Y\), shew that the equation of the asymptotes is \[ bX^2 - 2hXY + aY^2 = 0; \] also find the equation of the axes.
If a circle and a parabola intersect in four points, prove that their common chords are equally inclined to the axis of the parabola, and that the centre of mean position of the four points lies on the axis.
If the cross ratios of the two ranges \(PQRS\) and \(PXYZ\), having the point \(P\) in common, are equal, prove that the lines \(QX, RY, SZ\) are concurrent. \(AB\) and \(CD\) meet in \(U\), \(AC\) and \(BD\) in \(V\), \(UV\) cuts \(AD\) and \(BC\) in \(F\) and \(G\), and \(BF\) cuts \(AC\) in \(L\). Prove that the four points on the line \(AC\) form a harmonic range, and that \(LG, CF\) and \(AU\) are concurrent.
\(SY\) and \(SZ\) are the perpendiculars from the focus \(S\) of an ellipse on the tangent and normal at \(P\). Prove that \(YZ\) passes through the centre \(C\), and that \(CY\) is equal to half the major axis.
Prove that the reciprocal of a circle is a conic with a focus at the origin of reciprocation. Prove that a chord of a rectangular hyperbola which subtends a right angle at a focus touches a fixed parabola.