Problems

Prove that each of the pairs of lines \(ax^2 + 2hxy + by^2 = 0\), \(px^2 + 2qxy + ry^2 = 0\) is harmonically separated by the pair of lines \[ \begin{vmatrix} ax+hy & hx+by \\ px+qy & qx+ry \end{vmatrix} = 0. \] Shew that \((a-b)(p-r)+4hq=0\) is the necessary and sufficient condition that one of the angle bisectors of \(ax^2+2hxy+by^2=0\) should make an angle of \(\pi/4\) with one of the angle bisectors of \(px^2+2qxy+ry^2=0\), when the coordinate axes are perpendicular.

The coordinates of the vertices of a triangle referred to rectangular axes are \((R \cos\alpha, R\sin\alpha)\), \((R\cos\beta, R\sin\beta)\), \((R\cos\gamma, R\sin\gamma)\); find the coordinates of (i) the circumcentre, (ii) the centroid, (iii) the orthocentre.

The common points of the two rectangular hyperbolas \begin{align*} ax^2 + 2hxy - ay^2 + px + qy &= 0, \\ 2h'xy + p'x + q'y &= 0 \end{align*} are at the origin of coordinates and at the points \(L, M, N\); prove that the equation of the circle through \(L, M, N\) is \[ (ax+hy+p)(h'x+q') - (hx-ay+q)(h'y+p') = 0. \]

Shew that the foci of the central conic \(ax^2 + 2hxy + by^2 + c = 0\) are given by \[ \frac{x^2-y^2}{a-b} = \frac{xy}{h} = \frac{c}{ab-h^2}, \] the coordinate axes being rectangular. \par Find the coordinates of the real foci of the conic \[ 5x^2+4xy+2y^2-42x-24y+105=0. \]

The lines joining the vertices of a triangle \(XYZ\) to a point \(P\) cut the opposite sides in \(L, M, N\) and \(s\) is any conic through the four points \(P, L, M, N\). If this conic \(s\) cuts \(YZ\) again at \(A\) and if \(A'\) is the harmonic conjugate of \(A\) with respect to \(Y, Z\), prove that \(A'P\) is the tangent to \(s\) at \(P\).

Prove that \[ \left( \sum_{n=1}^N a_n b_n \right)^2 \le \sum_{n=1}^N a_n^2 \sum_{n=1}^N b_n^2, \] where \(a_1, \dots, a_N, b_1, \dots, b_N\) are positive. \par Hence or otherwise, prove that, for all real values of \(\rho\) and \(\sigma\), \[ \left( \sum_{n=1}^N c_n^\rho \right)^2 \le \sum_{n=1}^N c_n^{\rho-\sigma} \sum_{n=1}^N c_n^{\rho+\sigma}, \] where \(c_1, \dots, c_N\) are positive. \par Hence or otherwise, prove that \[ \sum_{n=1}^N a_n^2 \sum_{n=1}^N b_n^2 \le \sum_{n=1}^N \frac{a_n^3}{b_n} \sum_{n=1}^N a_n b_n. \]

Prove that \[ (x^2-1) \prod_{\nu=1}^{n-1} \left( x^2 - 2x \cos\frac{\pi\nu}{n} + 1 \right) = x^{2n}-1. \] Hence or otherwise, prove that \[ \prod_{\nu=1}^{\frac{1}{2}(n-1)} \cos\frac{\pi\nu}{n} = 2^{\frac{1}{2}(1-n)} \quad \text{for } n=3,5,7,\dots, \] and \[ \prod_{\nu=1}^{\frac{1}{2}n-1} \cos\frac{\pi\nu}{n} = \sqrt{n} \, 2^{1-n} \quad \text{for } n=4,6,8,\dots. \]

The functions \(\phi(x)\) and \(\psi(x)\) are differentiable in the interval \(a < x < b\); and \(\psi'(x ) > 0\) for \(a < x < b\). Prove that there is at least one number \(\xi\) between \(a\) and \(b\) such that \[ \frac{\phi(\xi)-\phi(a)}{\psi(b)-\psi(\xi)} = \frac{\phi'(\xi)}{\psi'(\xi)}. \] If \(\phi(x)=x^2\) and \(\psi(x)=x\), find a value of \(\xi\) in terms of \(a\) and \(b\).

Find the coefficients in the polynomial \(f_n(x)\) defined by \(f_n(x) = e^{-x} \frac{d^n}{dx^n} (x^n e^x)\). \par Prove the following identities, where \(n\) is a positive integer: \begin{align*} f_{n+1}(x) - (2n+1+x)f_n(x) + n^2 f_{n-1}(x) &= 0; \\ x f_n''(x) + (1+x)f_n'(x) - n f_n(x) &= 0. \end{align*}

Through each point \(P\) of a parabola with focus \(S\) a line \(PQ\) is drawn parallel to a fixed direction (in a definite sense) and of length equal to \(PS\). Show that \(Q\) describes a second parabola whose axis passes through a fixed point independent of the direction chosen.