From \(Q\) the middle point of a chord \(PP'\) of an ellipse, focus \(S\), \(QG\) is drawn perpendicular to \(PP'\) to meet the major axis in \(G\); prove that \[ 2SG = e(SP+SP'). \]
A circle touches the conic \(\frac{l}{r}=1+e\cos\theta\) at the point where \(\theta=\alpha\), and passes through the pole; prove that the two points of intersection with the conic lie on the straight line \[ \frac{l}{r}(1+2e\cos\alpha+e^2) = e\cos\theta+e^2\cos(\theta+\alpha). \]
If the four faces of a tetrahedron are equal in area, prove that they are equal in all respects.
If \[ \tan\phi = \frac{\sin\alpha\sin\theta}{\cos\theta-\cos\alpha}, \] prove that \[ \tan\theta = \frac{\sin\alpha\sin\phi}{\cos\phi\pm\cos\alpha}. \]
If \(p_n/q_n\) be the \(n\)th convergent to \(\sqrt{a^2+1}\) when expressed as a continued fraction, prove that \begin{align*} 2p_n &= q_{n-1}+q_{n+1} \\ \text{and} \quad 2(a^2+1)q_n &= p_{n-1}+p_{n+1}. \end{align*}
If \(x, y, z\) be real, prove that \[ a^2(x-y)(x-z)+b^2(y-x)(y-z)+c^2(z-x)(z-y) \] is always positive, provided that any two of the numbers \(a, b, c\) are together greater than the third.
If \(\sqrt{\frac{a}{x-a}} + \sqrt{\frac{b}{x-b}} + \sqrt{\frac{c}{x-c}} = \sqrt{\frac{abc}{(x-a)(x-b)(x-c)}}\), prove that \[ \frac{4abc}{x} = 2bc+2ca+2ab-a^2-b^2-c^2. \]
Three equal rods \(AB, BC, CD\) mutually at right angles are suspended by the end \(A\). Shew that the cosines of their inclinations to the vertical are as \(5:3:1\).
An elliptical cylinder rests with its curved surface in contact with two smooth planes each inclined at an angle of \(45^\circ\) to the horizontal; find the positions of equilibrium, and discuss their stability.
Two particles of masses \(m, m'\) connected by a light rod of length \(a+b\) are moving on a smooth horizontal plane in a direction perpendicular to the rod with uniform velocity \(v\); if the rod strikes a small fixed inelastic obstacle at a point whose distances from the masses are \(a, b\) respectively, prove that the measure of the blow is \(mm'(a+b)^2v/(ma^2+m'b^2)\).