Given the point \(P(ap^2, 2ap)\) on the parabola \(y^2 = 4ax\), prove that there are two circles which touch the parabola at \(P\) and also touch the \(x\)-axis \((y = 0)\). Verify that the middle point of the line joining the centres of these circles is at the same distance as \(P\) from the \(y\)-axis, but on the opposite side of it.
A point \(C\) is taken on the tangent to the rectangular hyperbola \(xy = k^2\) at its vertex \(A(k, k)\). The circle of centre \(C\) and radius \(CA\) cuts the hyperbola in four real points. Prove that there are just two parabolas through these four points and that one of them passes through the centre of the hyperbola while the other passes through the centre of the circle.
A conic \(S\) and two points \(U\), \(V\) not on it are given. A correspondence between two points \(P\), \(Q\) on the conic is set up as follows: \(UP\) meets the conic again in \(L\), \(VL\) meets the conic again in \(Q\). Determine the conditions under which the line \(PQ\) passes, for all positions of \(P\), through a fixed point \(W\) (to be identified).
A point \(P\) lies in the plane of a given triangle \(XYZ\). The lines \(XP\), \(YP\), \(ZP\) meet \(YZ\), \(ZX\), \(XY\) respectively in \(U\), \(V\), \(W\) and meet \(VW\), \(WU\), \(UV\) respectively in \(L\), \(M\), \(N\). Prove that \(VW\) meets \(MN\) on \(YZ\). The line \(VW\) is constrained to pass through a fixed point \(K\). Find the locus of (i) \(P\), (ii) \(L\). Discuss the particular cases that arise when \(K\) is on (a) \(YZ\), (b) \(XY\).
A triangle is to be circumscribed around a given circle. Prove that, if it is to have the minimum area, it must be equilateral.
Expand in a power series in \(x\), as far as the term in \(x^3\), $$e \log \log(e + x) - x e^{-x/e},$$ where \(e\) is the base of natural logarithms.
Prove that the curve given by \(x^y = y^x\) in the region \(x > 0\), \(y > 0\) of the Cartesian plane has just two branches, and sketch them. What are the coordinates of the point where they cross?
A curve is specified by its Cartesian coordinates \(x(t)\), \(y(t)\). \(s(t)\) is the arc-length along the curve, \(\psi(t)\) the angle between the tangent to the curve at the point \(t\) and the \(x\)-axis, \(R(t)\) the radius of curvature and \(X(t)\), \(Y(t)\) are the coordinates of the centre of curvature. Find equations enabling \(s\), \(\psi\), \(R\), \(X\), \(Y\) to be calculated. Defining also \(S(t)\) as the arc-length along the curve \((X(t), Y(t))\), show that \(|dS| = |dR|\), where \(dS\), \(dR\) are the infinitesimal changes in \(S\) and \(R\) corresponding to a change \(dt\) in the parameter \(t\). Interpret this result geometrically.
A set of functions \(y_n(x)\), \((n = 0, 1, 2, \ldots)\) is defined by $$y_n(x) = \cos(n \cos^{-1} x).$$ Show that
Prove that $$\int_1^n \log x \, dx < \sum_{r=2}^n \log r < \int_1^n \log x \, dx + \log n.$$ Hence, or otherwise, evaluate $$\lim_{n \to \infty} \frac{(n!)^{1/n}}{n}.$$