A common tangent to two non-intersecting circles \(C_1, C_2\) touches them at \(P_1, P_2\) respectively. \(L\) is one of the limiting points of the coaxal system determined by \(C_1, C_2\). \(P_1L\) meets \(C_1\) again at \(Q_1\) and \(P_2L\) meets \(C_2\) again at \(Q_2\). By inversion with respect to \(L\), or otherwise, prove that \(Q_1Q_2\) is a common tangent to \(C_1, C_2\).
A variable point \(P\) is taken on the parabola \(y^2 = a(x-a)\). The circle on the line joining \(P\) to the origin \(O\) as diameter meets the parabola \(y^2=4ax\) in three points besides \(O\). Prove that the normals at these three points are concurrent and their point of intersection lies on the parabola \[ y^2=4a(x+a). \]
Define carefully what you mean by an asymptote of a curve, and from your definition find the asymptotes of the hyperbola \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1. \] Find the asymptotes of the curve given in terms of the parameter \(t\) by the equations \[ x = \frac{h}{t^2-1}, \quad y = \frac{kt}{t^2-1}, \] where \(h\) and \(k\) are constants.
Three points \(A, B, C\) are given on a line \(l\). A fourth point \(D_1\) of the line is determined by the following construction. In any plane through \(l\) draw through \(A, B, C\) lines \(AQ_1R_1, BR_1P_1, CP_1Q_1\) intersecting in pairs in the points \(P_1, Q_1, R_1\). Let \(AP_1, BQ_1\) cut in \(U_1\). Then \(D_1\) is the point of intersection of \(R_1U_1\) with \(l\). Prove that, if the same construction is carried out with the lines \(AQ_2R_2, BR_2P_2, CP_2Q_2\), etc., leading to a point \(D_2\), then \(D_1\) and \(D_2\) are the same point.
\(ABCD\) is a plane quadrilateral. The line through \(A\) parallel to \(BC\) meets \(BD\) in \(P\), and the line through \(B\) parallel to \(AD\) meets \(AC\) in \(Q\). Prove that \(PQ\) is parallel to \(DC\).
A quadrilateral \(ABCD\) varies in such a manner that it is always inscribed in a fixed circle, of centre \(O\), and that the diagonals \(AC, BD\) intersect at right angles in a fixed point \(P\). Prove that the feet of the perpendiculars from \(O\) and \(P\) on to the sides of the quadrilateral all lie on a fixed circle.
By inversion, or otherwise, prove that, if \(A, B, C, D\) are four coplanar points, then the sum of any two of the products \(BC \cdot AD\), \(CA \cdot BD\), \(AB \cdot CD\) is greater than the third, unless the four points are concyclic. \(ABC\) is a triangle of which no one of the angles is greater than 120\(^\circ\). \(D\) is the vertex of the equilateral triangle described on \(BC\) on the side remote from \(A\); \(AD\) meets the circumcircle of \(BCD\) in \(O\). Prove that the sum of the distances \(PA, PB, PC\) of a point \(P\) of the plane from the vertices \(A, B, C\) is least when \(P\) is at \(O\). If the angle \(A\) is greater than 120\(^\circ\), what is the point which gives the least sum of distances from \(A, B, C\)?
A sphere passes through a fixed point \(P\) and touches two fixed planes. Prove that the locus of each point of contact is a circle, the centres of the circles lying on the perpendicular from \(P\) to the plane bisecting the angle between the given planes in which \(P\) lies. Prove also that the locus of the centre of the sphere is an ellipse.
Defining an involution on a straight line as a symmetrical bilinear relation \[ axx'+b(x+x')+c=0 \] between the distances \(x, x'\), from a fixed origin on the line, of two points \(P, P'\), establish the existence of a centre \(O\) on the line such that \(OP \cdot OP' = \text{const}\). A variable line through a fixed point \(K\) meets a given circle in points \(P, P'\). The joins of \(P, P'\) to a fixed point \(A\) on the circle meet a fixed chord through \(K\) in \(Q, Q'\). Prove that the points \(Q, Q'\) are pairs of an involution on the line. The line through \(A\) parallel to \(KQQ'\) meets the circle again in \(Y\), and \(KY\) meets the circle again in \(Z\). Prove that the four points \(A, Z, Q, Q'\) are concyclic.
Prove that a circle can be drawn through the four points of intersection of two parabolas whose axes are at right angles. Show that the point of intersection of the axes of the two parabolas bisects the join of the centre of this circle to that of the rectangular hyperbola through the same four points.