Prove that if two rectangular hyperbolas can be drawn through four points, then every conic through these points is a rectangular hyperbola. Hence, or otherwise, show that the orthocentre H of a triangle inscribed in a rectangular hyperbola lies on the curve. Show further, that if D is the fourth point in which the circumcircle of the triangle meets the curve, then DH passes through the centre of the rectangular hyperbola.
Prove that the ratio of the intercepts PG, PH on a normal at a point P to a central conic between P and the points G and H where the normal cuts the two principal axes is constant. Prove that for a point on any one of a system of confocal conics the product of the intercept on the normal between the two principal axes and the length of the projection on the normal of the radius from the centre of the conic to the point is constant.
Prove that pairs of tangent rays drawn from a point L to conics touching the sides of a quadrilateral are in involution, and that the joins of L to opposite pairs of vertices of the quadrilateral are members of the same involution. Hence, or otherwise, prove that if two pairs of opposite vertices of a quadrilateral are conjugate with respect to a conic S, the third pair of opposite vertices is also conjugate with respect to S.
Prove that in general two conics have one and only one common self conjugate triangle. Prove that if two quadrilaterals have the same diagonal triangle, the eight sides of the two quadrilaterals touch a conic.
The two diagonals AC and BD of a plane quadrilateral meet in O. Prove that \[ \text{area } \triangle\text{AOB} \times \text{area quadrilateral ABCD} = \text{area } \triangle\text{ABC} \times \text{area } \triangle\text{ABD}. \]
(i) Prove that if \(n\) is an odd integer, \(\sin n\theta + \cos n\theta\) regarded as a rational integral function of \(\sin\theta\) and \(\cos\theta\) is divisible either by \(\sin\theta+\cos\theta\), or by \(\sin\theta-\cos\theta\). (ii) Prove that if \(m\) and \(n\) are two different odd integers, or two different even integers, \(m\sin n\theta - n\sin m\theta\) is divisible by \(\sin^3\theta\).
The curve \(y=ax+bx^3\) passes through the points \((-0.2, 0.0167)\) and \((0.25, 0.026)\). Prove that the tangent at the origin makes an angle of approximately 34 seconds with the \(x\)-axis. Find the radius of curvature at the origin correct to six significant figures.
Prove that the function \[ \frac{1-\cos x}{\sin(x-a)} \quad (0 < a < \pi), \] has infinitely many maxima equal to 0 and minima equal to \(2\sin a\). Sketch the graph of the function.
Sketch the locus (the cycloid) given by \[ x=a(\theta+\sin\theta), \quad y=a(1+\cos\theta), \] for values of the parameter \(\theta\) between \(0\) and \(4\pi\). Prove that the normals to this curve all touch an equal cycloid, and draw this second curve in your diagram.
Discuss the convergence of the series \[ 1+z+z^2+\dots+z^n+\dots, \] where \(z\) may be real or complex.