Problems

The four common points of the parabola given by \(y^2-4ax=0\) and a rectangular hyperbola are all coincident at the point \(P\); prove that the centre of the rectangular hyperbola is the reflexion of \(P\) in the directrix of the parabola and that the asymptotes of the rectangular hyperbola are parallel to the bisectors of the angles between the tangent at \(P\) and the axis of the parabola.

The equation of a conic in homogeneous coordinates is \(s \equiv ax^2+by^2+cz^2=0\), where \(a+b+c=0\); if \(P(f,g,h)\) is a point on this conic \(s\), prove that the conic given by \(afyz+bgzx+chxy=0\) cuts \(s\) in four points \(P, P_1, P_2, P_3\) and that, if \((x_1, y_1, z_1)\) are the coordinates of \(P_1\), the equation of the common chord \(PP_1\) is \(ay_1z_1x + bz_1x_1y+cx_1y_1z=0\). Deduce that the equation of the other common chord \(P_2P_3\) is \(x_1x+y_1y+z_1z=0\) and that the triangle \(P_1P_2P_3\) is self polar with respect to the conic given by \(x^2+y^2+z^2=0\).

Shew that a straight tube whose cross-section is a regular hexagon can be completely blocked by a solid cube, and determine the ratio of an edge of the cube to a side of the hexagon.

Two conics inscribed in a triangle \(ABC\) touch \(BC\) at the same point \(P\), touch \(CA\) at \(Q, Q'\) and \(AB\) at \(R, R'\). The conics intersect in \(D, E\). \(PD\) meets \(CA, AB\) in \(L, M\). \(PE\) meets \(CA, AB\) in \(M', L'\). Prove that \(QR, Q'R', LL', MM', DE\) and \(BC\) are concurrent.

(i) Prove that, if \(a,b,c,d\) are real positive numbers not all equal, \[ 64(a^4+b^4+c^4+d^4) > (a+b+c+d)^4. \] (ii) Prove that, if \(m, n\) are positive integers and \(x, y\) and \(z\) real and positive and \(z^m = x^m+y^m\), then \(z^n >\) or \(< x^n+y^n\) according as \(n >\) or \(< m\).

Prove that \[ \tan^2\frac{\pi}{14} + \tan^2\frac{3\pi}{14} + \tan^2\frac{5\pi}{14} = 5, \] and \[ \cos^2\frac{\pi}{14} \cos^2\frac{3\pi}{14} \cos^2\frac{5\pi}{14} = \frac{7}{64}. \]

Prove that, if \(x,y,z\) are functions of two variables \(u,v\) given by the relations \[ x=f(u,v), \quad y=g(u,v), \quad z=h(u,v), \] then \[ \frac{\partial z}{\partial x} (f_u g_v - f_v g_u) = h_u g_v - h_v g_u \] and \[ \frac{\partial^2 z}{\partial x^2} (f_u g_v - f_v g_u)^3 = \begin{vmatrix} f_u & f_v & f_{uu}g_v^2 - 2f_{uv}g_u g_v + f_{vv} g_u^2 \\ g_u & g_v & g_{uu}g_v^2 - 2g_{uv}g_u g_v + g_{vv} g_u^2 \\ h_u & h_v & h_{uu}g_v^2 - 2h_{uv}g_u g_v + h_{vv} g_u^2 \end{vmatrix}, \] where suffixes denote partial differentiation.

Evaluate \[ \int_a^b \sqrt{\{(b-x)/(x-a)\}} dx, \quad a