Prove that the curve of intersection of two spheres is a circle in a plane perpendicular to the line that passes through their centres. Prove that the mid-points of the edges of a cube that do not pass through either end of a given diagonal are the vertices of a regular hexagon which lies in the plane which bisects the given diagonal at right angles.
Prove that the circumcircle of the triangle formed by three tangents to a parabola passes through the focus. The tangents to a parabola at \(P\) and \(Q\) meet in \(T\), and the tangent that is parallel to \(PQ\) cuts \(TP, TQ\) in \(Y, Z\). Prove that, if the circle \(TYZ\) cuts the diameter through \(T\) in \(H\), then the line joining \(H\) to the focus is parallel to \(PQ\).
Prove that, if any chord \(PQ\) of a hyperbola cuts the asymptotes in \(M, N\), then \(MP = QN\). Having given three points \(A, B, C\) on a hyperbola and one asymptote, shew how to construct (1) the other asymptote, and (2) the tangent at \(A\).
Prove that \(a+b-c-d\) is a factor of \[ (a+b+c+d)^3 - 6(a+b+c+d)(a^2+b^2+c^2+d^2) + 8(a^3+b^3+c^3+d^3); \] and resolve this expression into its simple factors.
Prove that, if \(s_n = \alpha^n+\beta^n\), where \(\alpha, \beta\) are the roots of \(x^2-ax+b=0\), \[ s_n = a s_{n-1} - b s_{n-2}. \] Hence, or otherwise, prove that if \(u_n\) denotes the expression \[ a^n s_n + n b a^{n-1} s_{n-1} + \frac{n(n+1)}{1.2} b^2 a^{n-2} s_{n-2} + \dots + \frac{(2n-2)!}{(n-1)!(n-1)!} b^{n-1} a s_1, \] then \[ u_n = a^n u_{n-1} = a^{2n}. \]
Prove that, when \(b-a\) is small compared with \(a\) the expression \(\log_e(b/a)\) is approximately equivalent to \(2(b-a)/(b+a)\). Shew that, if \(b-a\) does not exceed \(a/10\), the error in using this approximation is less than \(.0001\).
A weight of 20 oz. is supported by two strings one of which is tied to a fixed point \(A\) while the other passes over a smooth peg at \(B\) in the same horizontal line as \(A\) and has its other end attached to a weight of 7 oz. If the length of the former string and the distance \(AB\) are each 10 inches, determine the depth of the 20 oz. weight below \(AB\) in the position of equilibrium.
A piece of uniform wire is bent into the shape of an isosceles triangle \(ABC\) in which \(AB=AC\). The triangle hangs in a vertical plane with \(BC\) in contact with a rough peg. Shew that the triangle will rest in equilibrium whatever point of \(BC\) is in contact with the peg provided that the coefficient of friction \(> 2\tan\frac{A}{2}(1+\sin\frac{A}{2})\).
A motor bicycle with side car weighing 3 cwt. attains a speed of 20 miles per hour when running down an incline of 1 in 20 without the use of the engine. But up the same incline the greatest speed that can be attained is 40 miles per hour. Assuming that the resistance varies as the square of the velocity, determine the horse-power developed by the engine.
Find the direction in which a particle must be projected from a point with given velocity in order that the range on an inclined plane through the point may be a maximum. Prove that, if the difference in level between two points \(A\) and \(B\) is \(L\), the velocity of projection from \(A\) in order that \(B\) may be just within range of \(A\) is \(\sqrt{\{g(AB \pm L)\}}\) according as \(B\) is above or below \(A\).