Parallel lines \(LL', MM'\) are drawn in a fixed direction at a constant distance apart to meet two fixed lines \(OX, OY\) in \(L\) and \(L'\), \(M\) and \(M'\) respectively. Prove that the envelope of \(LM'\) is a parabola touching \(OX, OY\), and find the points of contact with \(OX, OY\).
A triangle is self-conjugate with regard to the conic \(ax^2+by^2=1\), and the coordinates of its orthocentre are \((f, g)\). Shew that the vertices lie on the hyperbola \[ bfy - agx + (a-b)xy = 0, \] and that the sides touch the parabola \[ \sqrt{fx} + \sqrt{-gy} = \sqrt{(1/b - 1/a)}. \]
Prove that the cone joining any point to a circular section of a sphere cuts the sphere again in a circle.
Prove that, if \(bc+ca+ab=0\), then \[ \Sigma a^5 = \Sigma(a^2)\{\Sigma(a^3)+2abc\}. \]
Prove that, if \(x\) is large, \[ \left(1+\frac{1}{x}\right)^{x+\frac{1}{2}} = e\left(1+\frac{1}{12x^2}+\dots\right). \]
Prove that, if \(x\) denote any convergent of the continued fraction \[ \frac{1}{a+} \frac{1}{b+} \frac{1}{a+} \frac{1}{b+} \dots, \] then the next convergent is \(a+a/bx\).
Prove that, if the incircle of a triangle passes through the circumcentre, then \[ \cos A + \cos B + \cos C = \sqrt{2}. \]
A uniform heavy beam rests across and at right angles to two horizontal rails which support the beam at its points of trisection. Shew that, if the coefficient of friction is the same at both contacts and a gradually increasing force is applied to the beam at one end parallel to the rails, equilibrium will be broken by sliding on the nearer rail only.
A uniform hemisphere of weight \(W\) and radius \(a\) is placed symmetrically on top of a fixed sphere of radius \(b\), the curved surfaces being in contact and sufficiently rough to prevent sliding. Shew that if the hemisphere is rolled through a small angle \(\theta\) the gain of potential energy is approximately \[ \frac{1}{2}W(3b-5a)a\theta^2/(a+b), \] and deduce the condition for stability.
A weight of 200 lb. hanging from a rope is raised by a force which starts at 300 lb. and decreases uniformly by 1 lb. for every foot the weight is lifted. Find the velocity when the weight has risen 40 feet.