The normal at the point \(P\) on the parabola \(y^2=4ax\) meets the parabola again in \(Q\), and \(R\) is the pole of \(PQ\). The chord through \(P\) and the focus \(S\) meets the parabola again in \(T\). Prove that \(RT\) is parallel to the axis of the parabola, and also that \(PR\) is bisected by the directrix. Find the locus of \(R\) as \(P\) varies on the given parabola.
The polar of the point \(P\) with respect to the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] meets the ellipse in the points \(Q\) and \(R\). Given that \(QR\) is of constant length \(2c\), prove that the locus of \(P\) is \[ \left(\frac{x^2}{a^2} + \frac{y^2}{b^2} - 1\right)\left(\frac{a^2y^2}{b^4} + \frac{b^2x^2}{a^4}\right) = c^2 \left(\frac{x^2}{a^4} + \frac{y^2}{b^4}\right). \]
Prove that, in general, one conic of the confocal system \[ \frac{x^2}{a+\lambda} + \frac{y^2}{b+\lambda} = 1, \] where \(\lambda\) is a parameter, touches the line \(x \cos \alpha + y \sin \alpha = p\), and find the co-ordinates of the point of contact. Find also the equation of the conic of the confocal system which has the given line as a normal.
Prove that the inverse of a circle is either a circle or a straight line. Prove also that the angle at which two curves cut is unaltered by inversion. Given a coaxial system of circles intersecting in two real points, prove that there is at least one circle of the system orthogonal to a given circle, and discuss the conditions that there should be more than one such circle of the system. Prove also that there are either none, one, or two real circles of the system which touch a given circle, and discuss the conditions for each case.
Prove Desargues' theorem, that if two triangles in the same plane are in perspective from a point then their corresponding sides intersect in three collinear points (the axis of perspective). Prove that if three triangles in the same plane are in perspective, then the three axes of perspective of the triangles taken in pairs meet in a point. (The converse of Desargues' Theorem may be assumed.)
Through the vertices \(A, B, C\) of an acute-angled triangle \(ABC\) straight lines \(VAW, WBU, UCV\) are drawn so that the triangles \(ABC, UVW\) are similar. Prove that the circles \(BCU, CAV, ABW\) are equal, and that they meet in a point which is the orthocentre of the triangle \(ABC\) and the circumcentre of the triangle \(UVW\).
State and prove the theorem of Menelaus for a transversal \(LMN\) of a triangle \(ABC\). \(ABCD\) is a given parallelogram; points \(Q, V\) are taken on \(AD, BC\) respectively so that \(QV\) is parallel to \(AB\), and points \(R, W\) are taken on \(AB, CD\) respectively so that \(RW\) is parallel to \(AD\). Prove that \(QR\) meets \(VW\) on the diagonal \(BD\).
Define a hyperbola, and prove that, if \(A, B\) are two given points, then the locus of a point \(P\) which moves so that \(PA-PB\) is constant is one branch of a hyperbola. Two given non-intersecting circles have centres \(A, B\) and radii \(a, b\) respectively (\(a>b\)). \(P\) is the centre of a circle touching each of the given circles, internally or externally. Find the complete locus of \(P\), distinguishing the different cases.
A tetrahedron \(ABCD\) has the property that a sphere can be drawn to touch each of its six edges. Prove that \[ AD+BC=BD+CA=CD+AB. \] Investigate whether the converse result is true that, if a tetrahedron \(ABCD\) has the property that \[ AD+BC=BD+CA=CD+AB, \] then there exists a sphere touching each of its six edges.
The coordinates of a variable point \(T\) of a certain curve are given in terms of a parameter \(t\) by means of the relations \[ x=at^3, \quad y=at, \] where \(a\) is constant. Prove that, if \(P, Q, R\) are three distinct collinear points of the curve, with parameters \(p, q, r\), then \[ p+q+r=0. \] Prove also the converse result that, if \(p+q+r=0\), then the points are collinear. \(A, B, C\) are three points on the curve. The lines \(BC, CA, AB\) meet the curve again in \(L, M, N\) and the lines joining \(A, B, C\) to the origin meet the curve again in \(U, V, W\). Prove that \(LU, MV, NW\) are concurrent.