Problems

A series of circles touch a given straight line at a given point. Show that the middle points of the chords of contact of the tangents from a fixed point all lie on a circle.

State and prove the property from which the nine points circle of a triangle derives its name. \(RS\) and \(PQ\) are the diameters of the nine-points circle and the circumcircle of the triangle \(ABC\) which are at right angles to the side \(BC\) and \(E\) is the middle point of \(BC\); prove that the lines \(ER, ES, AP, AQ\) form a rectangle.

Define the polar of a point with respect to a circle and show that a straight line through a point cutting a circle is divided harmonically by the circle, the point and the polar of the point. \(TP, TQ\) are tangents to a circle. The perpendicular from \(T\) on any diameter \(AB\) cuts that diameter in \(X\). Prove that \(PX.XQ=AX.XB\).

Prove that the polar reciprocal of a circle with respect to another circle is a conic and find the positions of the foci of the conic. If three tangents and a point on the director circle of a conic are given, prove that the conic touches a fourth fixed line.

Prove that any chord of a rectangular hyperbola subtends at the ends of a diameter angles which are either equal or supplementary. Show that the angle between two tangents to a rectangular hyperbola is equal or supplementary to the angle which their chord of contact subtends at the centre, and that the bisectors of these angles meet on the chord of contact.

Find the condition that the line \(lx+my+n=0\) may touch the circle \[ (x-a)^2+(y-b)^2=r^2. \] Show that two circles of the system \(x^2+y^2+2\theta x-c^2=0\), where \(\theta\) is a variable parameter, may be drawn to touch a straight line, and that if the circles cut orthogonally, the straight line touches the ellipse \(2x^2+y^2=2c^2\).

Find the equation of the chord joining the points \((at^2, 2at)(at'^2, 2at')\) on the parabola \(y^2=4ax\). A circle is drawn through the two points where the straight line \(lx+my+n=0\) cuts the parabola \(y^2=4ax\) to touch the parabola at \(P\). Find the coordinates of \(P\) and also those of the centre of the circle.

Find the equation of the normal at the point on the ellipse \(x^2/a^2+y^2/b^2=1\) whose eccentric angle is \(\phi\). If the normals at two points on this ellipse intersect on the ellipse, prove that the tangents at the two points intersect on \[ (x^2/a^2+y^2/b^2)^2 = (a^2-b^2)^2\{(a^2-x^2)^2y^2/b^4 + (b^2-y^2)^2x^2/a^4\}/a^4b^4. \]

Interpret the locus \(S-kL^2=0\), where \(S=0\) is a conic and \(L=0\) is a straight line. A circle has double contact with an ellipse at \(Q\) and \(Q'\); prove that the length of the tangent to the circle from any point \(P\) on the ellipse is \(SP-SQ\) where \(S\) is a focus.

Find the condition that the lines \(l\alpha+m\beta+n\gamma=0\), \(l'\alpha+m'\beta+n'\gamma=0\) in trilinear coordinates may be parallel. The lines \(l\alpha+m\beta+n\gamma=0\), \(l'\alpha+m'\beta+n'\gamma=0\) meet the sides \(BC, CA, AB\) of the fundamental triangle in \(D, E, F\) and \(D', E', F'\) respectively. \(D_1, D_1'\) are the harmonic conjugates of \(D, D'\) respectively with regard to \(B\) and \(C\). Similarly points \(E_1, E_1'\) and \(F_1, F_1'\) are found on \(CA\) and \(AB\). Prove that the lines \(AD_1, BE_1, CF_1\) intersect in a point \(P\) and \(AD_1', BE_1', CF_1'\) intersect in \(P'\) and find the equation of the line \(PP'\).