Problems

Interpret the equation \[ S + \lambda uv = 0, \] where \(S=0\) is the equation of a conic, \(u=0\) and \(v=0\) are the equations of two straight lines, and \(\lambda\) is a constant. Two conics \(S_1\) and \(S_2\) meet in four points, and a conic \(S\) has double contact with \(S_1\) and with \(S_2\). Prove that the point of intersection \(O\) of the two chords of contact is the point of intersection of one of the pairs of common chords of \(S_1\) and \(S_2\). Shew that \(O\) is also the meet of the diagonals \(AC, BD\) of the quadrilateral formed by the common tangents \(AB, CD\) of \(S\) and \(S_1\) and the common tangents \(BC, DA\) of \(S\) and \(S_2\).

D, E, F are the middle points of the sides BC, CA, AB respectively of the triangle ABC, X is any point in the plane of the triangle and AX, BX, CX meet the sides of the triangle at P, Q, R; prove that the lines joining D, E, F to the middle points of AP, BQ, CR respectively meet at a point K. Prove also that, if X is the orthocentre of the triangle ABC, the perpendicular distances of K from the sides of the triangle ABC are proportional to BC, CA, AB.

State (without proof) a construction for (i) the radical axis, (ii) the limiting points of a coaxal system of non-intersecting circles, when two circles of the system are given. I is the centre of the inscribed circle of the triangle ABC and the lines through I perpendicular to IA, IB, IC meet BC, CA, AB at the points P, Q, R respectively; prove that P, Q, R lie on a straight line, which is perpendicular to the line joining I to the circumcentre of the triangle ABC.

P is a variable point on a parabola with vertex A and focus S, and M, N are the feet of the perpendiculars from P to the tangent at the vertex and to the axis AS; prove that the envelope of MN is another parabola, and determine the positions of the focus and directrix of this parabola in relation to A, S.

Explain what is meant by two related (homographic) ranges of points (P, Q, R, \dots) and (P', Q', R', \dots) on a straight line, and shew that there are two self-corresponding points real, coincident or imaginary. If the two self-corresponding points are coincident at O, and Q coincides with P', prove that the pairs (O, P') and (P, Q') are harmonically conjugate and that a point V and a line l can be found so that, when the two ranges are projected from V upon l, the two resulting ranges have their corresponding segments equal.

A fixed plane p meets a fixed sphere in a small circle Y and a variable plane p' meets the sphere in a circle Y'; prove that, if the circles Y, Y' cut each other orthogonally, the plane p' passes through a fixed point P. Prove also that, if the plane p passes through a fixed line, the corresponding points P lie on another fixed line.

A variable straight line through the point \((x_1, y_1)\) meets the pair of lines \[ ax^2+2hxy+by^2=0 \] at the points P, Q; prove that the locus of the orthocentre of the triangle OPQ is the hyperbola \[ bx^2-2hxy+ay^2-(a+b)(x_1x+y_1y)=0. \]

Prove that the envelope of the radical axis of a fixed circle and a variable circle, which touches two fixed straight lines, is the pair of parabolas which pass through the common points of the fixed circle and the two fixed lines.

Prove that (i) the polar lines of the point \(P_1(x_1, y_1)\) with respect to the system of conics confocal with the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) touch a parabola, whose focus is at the point \(P_2(x_2,y_2)\), where \(x_1y_2+x_2y_1=0\), \(x_1x_2-y_1y_2=a^2-b^2\); (ii) this parabola touches the tangents at the feet of the four normals from \(P_1\) to any conic of the confocal system.

Prove that the parabola \((x-y)^2+8x-4y=0\) and the hyperbola \[ 16x^2-3y^2-32x+16y=0 \] touch each other, and find the equations of (i) the tangent at the point of contact, (ii) the other two common tangents.