Of two circles which cut orthogonally one has a fixed centre and the other passes through two fixed points. Shew that their radical axis passes through a fixed point, and determine its position.
Given an obtuse-angled triangle, determine a circle of which it is the self-conjugate triangle. Show that the circle is real and that there cannot be more than one.
Find the sum of the cubes of the first \(n\) natural numbers, and determine a set of \(2n+1\) consecutive integers such that their sum bears to the sum of their cubes the ratio \(1:m^2+n^2+n\).
Prove that in any triangle \(ABC\), \[ \cos A + \cos B + \cos C \le \frac{3}{2}, \] \[ \cot B \cot C + \cot C \cot A + \cot A \cot B \ge 9. \]
Find the equation of the axes of the conic given by the general equation. Trace roughly the curve \(15x^2+40xy+24y^2-120x-120y=0\), and prove that its foci lie on the axes of coordinates.
Prove that of the circles \begin{align*} b(x^2+y^2) + a^2(2y-b) &= 0, \\ a(x^2+y^2) + b^2(2x-a) &= 0, \\ ab(x^2+y^2) + 2c^2(bx-ay) &= 0, \end{align*} the first two intersect at an angle of 120\(^\circ\); also that the third, if it is coaxial with the others, intersects each of them at 120\(^\circ\).
Determine the radius of curvature at any point of a curve whose coordinates are given in terms of a single parameter \(\theta\). The normal at any point \(P\) of the curve \(x=a\cos^3\theta, y=a\sin^3\theta\) meets the circle \(x^2+y^2=a^2\) in the points \(Q, R\). Prove that \(RP=3PQ=\rho\) the radius of curvature at \(P\). Also that, if \(s=0\) when \(\theta=\frac{1}{4}\pi\), \(16s^2+4p^2=9a^2\).
Find the coordinates of the double point of the cubic whose equation is \[ xy(5x+y-6)+3x+3y-2=0. \] Write down the equation of the tangents at the double point. Are they real?
Prove that, if \(\cos\beta = \cos\theta\cos\phi+\sin\theta\sin\phi\cos\alpha\), and \(\sin\alpha = e\sin\beta\) \[ d\theta\{1-e^2\sin^2\phi\}^{\frac{1}{2}} + d\phi\{1-e^2\sin^2\theta\}^{\frac{1}{2}} = 0. \]
Integrate with respect to \(\theta\) the expressions \(\frac{1}{\sin^3\theta}\) and \(\frac{5}{1+2\cot\theta}\). Prove that the straight line \(2a^2x=9b^2y\) cuts off from the curve \(b^2y=x^2(a-x)\) two segments which are equal in area.