Problems

Prove that the expression \[5x^2 + 6y^2 + 7z^2 + 2yz + 4zx + 10xy\] is positive for all real values of \(x\), \(y\), \(z\), other than \(x = 0\), \(y = 0\), \(z = 0\). Find a set of real values of \(x\), \(y\), \(z\) for which the expression \[5x^2 + 6y^2 + 7z^2 - 2yz - 4zx - 10xy\] is negative. (If you quote a general test in support of your arguments, you must prove it.)

Prove that, if the simultaneous equations \begin{align} 3x + ky + 2z &= \lambda x,\\ kx + 3y + 2z &= \lambda y,\\ 2x + 2y + z &= \lambda z \end{align} have a solution in which \(x\), \(y\), \(z\) are not all zero, then \[(1-\lambda) k^2 - 8k + (\lambda + 1)(\lambda - 3)(\lambda - 5) = 0.\] When this condition is satisfied, find formulae for the most general solutions in the two cases (i) \(\lambda = 1\), (ii) \(\lambda = 3\).

Two circles intersect in distinct points \(A\), \(B\); a variable chord through \(A\) meets one circle again in \(P\) and the other in \(Q\). Find that position of the chord for which the length of \(PQ\) is greatest.

A cube of side \(2a\) has horizontal faces \(ABCD\), \(A'B'C'D'\) and vertical edges \(AA'\), \(BB'\), \(CC'\), \(DD'\). Find the radius of the sphere inscribed in the tetrahedron \(AB'CD'\).

Two perpendicular straight lines meet at \(O\); a circle of centre \(P\) cuts the first line in points \(A\), \(B\) and the second in points \(C\), \(D\). The point \(O\) does not lie on either of the segments \(AB\) or \(CD\). Prove that the segment \(OP\) has a point in common with each of the segments \(AC\), \(AD\), \(BC\), \(BD\). (By the segment \(XY\) is meant that part of the straight line \(XY\) which lies between \(X\) and \(Y\).)

The point \(P\) on the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] has eccentric angle \(\theta\), so that the coordinates of \(P\) are \((a\cos\theta, b\sin\theta)\). A circle is drawn concentric with the ellipse and passing through \(P\). Find expressions for

1. [(i)] the angle which the common tangents of the ellipse and the circle make with the \(x\)-axis.
2. [(ii)] the angle between the tangents at \(P\) to the ellipse and the circle.

The circle of radius \(3a\) with its centre at the focus \((a, 0)\) of the parabola \(y^2 = 4ax\) cuts the parabola at \(P\) and \(Q\), and \(O\) is the origin of coordinates. The tangent at \(P\) to the parabola cuts the \(x\)-axis in \(L\) and the \(y\)-axis in \(M\). Prove that \(PQ\) is the tangent at \(O\) to the circle \(LOM\). Prove also that \(O\) is the orthocentre of the triangle \(LPQ\).

Given a sheet of paper on which are drawn the hyperbola \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\] and its asymptotes, with the vertex \(A (a, 0)\) marked, show how to draw (with ungraduated ruler and compasses only) that chord \(AP\) which, meeting the asymptotes in \(L\), \(M\), is such that \(A\) and \(P\) are the points of trisection of the segment \(LM\). Justify your construction. Prove that, if \(U\) is the foot of the perpendicular from \(M\) on the \(x\)-axis (where \(L\), \(A\), \(P\), \(U\) is the origin of coordinates), then \(A\) is a point of trisection of \(OU\), where \(O\) is the origin of coordinates.

Three points \(A\), \(B\), \(C\) lie on a line \(l\) and three points \(P\), \(Q\), \(R\) lie on a line \(m\). Prove that

1. [(i)] if \(l\) and \(m\) are coplanar, the pairs of lines \((BR, CQ)\), \((CP, AR)\), \((AQ, BP)\) intersect in three points which are collinear;
2. [(ii)] if \(l\) and \(m\) are not coplanar, and if \(X\) is an arbitrary point in space, then the three transversals, one from \(X\) to each of the pairs of lines named in (i), are coplanar.

The homogeneous coordinates of a point \(U\) with respect to a triangle of reference \(P(\alpha, \beta, \gamma)\) are \((1, 1, 1)\). The lines \(XU\), \(YU\), \(ZU\) meet \(YZ\), \(ZX\), \(XY\) respectively in \(L\), \(M\), \(N\), and \(YXLLP\), \(ZLMUP\) meet the sides \(YZ\), \(ZX\), \(XY\) respectively in points all of which lie on a conic with respect to which the triangle \(XYZ\) is self-polar.