Problems

If a circle of radius R cuts a rectangular hyperbola whose centre is O at the points A, B, C, D, prove that \[ OA^2+OB^2+OC^2+OD^2 = 4R^2. \]

The equations of a conic and a line referred to rectangular axes are \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \quad \text{and} \quad lx+my+n=0; \] prove that the line is a principal axis of the conic, if \[ al+hm : hl+bm : gl+fm = l:m:n, \] and deduce that the equation of the principal axes is \[ h(u^2-v^2)+(b-a)uv = 0, \] where \(u \equiv ax+hy+g, v \equiv hx+by+f\). \par Interpret these results geometrically, when \(a=\alpha^2, h=\alpha\beta, b=\beta^2\).

If \(lx+my+1=0\) is the equation of a straight line referred to rectangular axes, interpret geometrically the constants \(a, b, a', b', c\) in relation to the conic whose tangential (envelope) equation is \[ (al+bm+1)(a'l+b'm+1) = c^2(l^2+m^2). \] If \(\Sigma'\) is any conic touching the two pairs of tangents from the points P, Q to a conic \(\Sigma\), prove that the four common tangents of any conic confocal with \(\Sigma\) and of any conic confocal with \(\Sigma'\) touch a conic with foci at P, Q.

If P, Q, R are three points with homogeneous coordinates \((p, g, h), (f, q, h), (f, g, r)\), respectively, and XYZ is the triangle of reference, find the equation of the line through the three intersections of the pairs of lines (QR, YZ), (RP, ZX), (PQ, XY). \par Shew also that, if \(fgh=pqr\), the lines XQ, YR, ZP are concurrent and the lines XR, YP, ZQ are concurrent.

A, B and C are three points in the plane of a conic S; the pole of BC with respect to S is A', the pole of CA is B', and the pole of AB is C'. Prove that AA', BB', CC' meet in a point P. \par If B and C are fixed, and A moves in a straight line \(l\), prove that the locus of P is a conic through B, C and A'. What happens if \(l\) passes through A'?

The base of a cone is bounded by a circle of radius \(a\) lying in a horizontal plane. The centre of the circle is O, and A is a fixed point on the circle. The vertex V of the cone lies in the vertical plane through OA, VO is of length \(b\) (greater than \(a\)), and VOA is an acute angle \(\alpha\). P is a variable point on the circle, and the angle AOP is denoted by \(\phi\). Prove that \[ \cos\text{VOP} = \cos\alpha\cos\phi. \] Discuss the maximum value of the angle between VO and the generator VP for different positions of P on the circle. Prove that, if \(b < a \sec\alpha\), the maximum is \(\beta\), where \[ \tan\beta = \frac{a\sin\alpha}{b-a\cos\alpha}, \] and that, if \(b>a\sec\alpha\), the maximum is \(\sin^{-1}(a/b)\).

The variables \(x,y\) are connected by the homographic relation \[ y = \frac{ax+b}{cx+d} \quad (c\neq 0, ad-bc \neq 0). \] Prove (i) that the cross-ratio of any four \(x\)'s is equal to the cross-ratio of the corresponding four \(y\)'s, (ii) that, if the self-corresponding points \(\alpha, \beta\) are distinct, \[ \frac{y-\alpha}{y-\beta} / \frac{x-\alpha}{x-\beta} \] has a constant value independent of \(x\). \par A sequence of numbers \(u_1, u_2, u_3, \dots\) is determined by the relations \[ u_1 = 2a, \quad u_{n+1} = 2a + \frac{b^2}{u_n} \quad (n\ge 1), \] where \(a, b\) are positive constants. Determine \(u_n\) in terms of \(n\) and the constants \(a, b\).

Express \((x^{2n+1}+1)/(x+1)\) as a product of real quadratic factors. \par If \(k\) is an odd integer greater than 1, prove that \[ \cos\frac{\pi}{k}\cos\frac{3\pi}{k}\cos\frac{5\pi}{k}\dots\cos\frac{(k-2)\pi}{k} = (-)^m 2^{-(k-1)/2}, \] where \(k\) is of the form \(4m+1\) or \(4m+3\); and that \[ \tan\frac{\pi}{2k}\tan\frac{3\pi}{2k}\tan\frac{5\pi}{2k}\dots\tan\frac{(k-2)\pi}{2k} = \frac{1}{\sqrt{k}}. \]

The function \(f(x)\) has a continuous second derivative \(f''(x)\) in the interval \([a,b]\); prove that, if \(a

If \(f(x)\) is continuous for all real values of \(x\), prove that \[ \frac{d}{dx}\int_0^x f(t)dt = f(x). \] The function \(\theta(x)\) has continuous derivatives \(\theta'(x), \theta''(x), \theta'''(x)\), for all \(x>0\), and \(\theta(x), \theta'(x)\) are positive. The functions \(\phi(x), \psi(x)\) are defined, for \(x>0\), by the equations \[ \phi(x) = \frac{1}{x}\int_0^x \theta'(t)dt, \quad \psi(x) = \theta(x)\phi(x). \] Prove that \(\phi(x), \psi(x)\) are positive when \(x\) is positive, and that \[ \frac{\psi''(x)}{\psi(x)} = \frac{\theta''(x)}{\theta(x)} + \frac{\phi''(x)}{\phi(x)}. \] If \(\theta'(x)\) tends to a positive limit as \(x\to\infty\), prove that \(\phi(x)/x^2 \to \frac{1}{2}\) as \(x\to\infty\).