Problems

Prove that any line cuts the sides of a triangle in segments the continued product of the ratios of which is minus unity. Extend this to the case of a quadrilateral \(ABCD\) cut by a plane, where \(A, B, C, D\) are not coplanar. If a sphere touches internally the four sides of such a quadrilateral, show that the four points of contact are coplanar.

Show that:

1. [(i)] A circle \(C\) and a pair of points inverse with respect to \(C\) invert into a circle \(C'\) and a pair of points inverse with respect to \(C'\).
2. [(ii)] The cross ratio of four collinear points is equal to the cross ratio of the pencil formed by joining the inverses of the four points to the inverse of any point on the line.

Prove that if \(S, H\) are the foci of an ellipse and \(SY, HZ\) are the perpendiculars from \(S, H\) on any tangent, \(Y, Z\) lie on a fixed circle and \(SY.HZ\) is constant. A variable line cuts two fixed circles, one at \(A, B\) and the other at \(C, D\), so that the range \((ACBD)\) is harmonic. Show that the line touches a fixed conic with the centres of the circles as foci.

Interpret geometrically the expression \(S\equiv(x-\alpha)^2+(y-\beta)^2-c^2\) with regard to the circle \(S=0\). Show that the locus of the centre of the circle of radius \(\kappa\) which cuts the circles \(S=0, S'=0\) at the same angle is \(Sc' - S'c = \kappa^2(c'-c)\).

Prove that the tangents at the ends of a focal chord of a parabola meet at right angles on the directrix. Prove that the circle circumscribing the triangle formed by a variable focal chord and the tangents at its extremities touches the directrix and also a fixed circle.

Find the equations of the tangent and normal at any point of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) in terms of the eccentric angle. From any point \(P\) of this ellipse, \(PM, PN\) are drawn perpendicular to the axes. Show that \(MN\) is normal to an ellipse whose equation is \(\frac{x^2}{a^6}+\frac{y^2}{b^6}=\frac{1}{(a^2-b^2)^2}\).

Find the equation of the pair of tangents from \((x',y')\) to the conic \(Ax^2+By^2=1\). Show that if the pairs of tangents from a point to \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) and to \(\frac{x^2}{a^2+\lambda}+\frac{y^2}{b^2+\lambda}=1\) form a harmonic pencil, the locus of the point is the conic \[ \frac{2a^2b^2}{a^2+b^2+\lambda}\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}-1\right)+\lambda(x^2+y^2-a^2-b^2)=0. \]

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 show that the directrix is \(ax+a'y+b^2+b'^2=0\).