The normal at \(P\) to a parabola whose focus is \(S\) cuts the axis in \(G\). Prove that the locus of the middle point of \(PG\) is a parabola whose vertex is \(S\).
If \(p\) and \(q\) are the roots of \[ \frac{1}{x+a} + \frac{1}{x+b} + \frac{1}{x} = 0, \] and \[ a^2+b^2 = 4ab, \] then \[ p^2+q^2 = 6pq. \]
Prove that \[ \frac{e-1}{e+1} + \frac{1}{3}\left(\frac{e-1}{e+1}\right)^3 + \frac{1}{5}\left(\frac{e-1}{e+1}\right)^5 + \dots = \frac{1}{2}, \] where \(e\) is the base of Napierian logarithms.
Prove that \(\sin(A+B+C)\) is one factor of \[ 1-\cos^2 2A - \cos^2 2B - \cos^2 2C + 2\cos 2A \cos 2B \cos 2C, \] and find three other factors of a similar form.
A statue on a pedestal stands on a slope of inclination \(\theta\), and at a certain point on the slope the statue subtends an angle \(\alpha\) and the pedestal subtends an angle \(\beta\). If the statue again subtends an angle \(\alpha\) at a point \(a\) feet nearer up the slope, show that the vertical heights of the statue and pedestal are \(a \dfrac{\sin\alpha}{\cos(\alpha+\theta+2\beta)}\) and \(a \dfrac{\sin\beta\cos(\alpha+\theta+\beta)}{\cos\theta\cos(\alpha+\theta+2\beta)}\) respectively.
\(O\) is the vertex of the parabola \(y^2=4ax\) and \(P,Q\) are the points in which it meets the line \(lx+my=4na\). Prove that the internal and external bisectors of the angle \(POQ\) are given by the equation \(m(x^2-y^2)=2(l+n)xy\).
A circle is drawn touching the ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\) at any point and passing through the centre. Show that the locus of the foot of the perpendicular from the centre to the common chord of the circle and the ellipse is the ellipse \(a^2x^2+b^2y^2=\dfrac{a^4b^4}{(a^2-b^2)^2}\).
Tangents are drawn to an hyperbola from points on a second hyperbola having the same asymptotes; prove that the chord of contact envelopes a third hyperbola having the same asymptotes.
Trace the curve \[ y^2(a+x) = x^2(a-x). \] Prove that the co-ordinates of any point on the curve can be expressed in the form \(x=a\cos 2\theta, y=a(\sin 2\theta - \tan\theta)\), and that the area of the loop of the curve is \(2a^2(1-\dfrac{\pi}{4})\).
A sphere of weight \(W\) and radius \(a\) rests on three equal rods of length \(2a\) which are pinned together at their ends to form an equilateral triangle. If the vertices of the triangle are supported in a horizontal plane and there is no friction, prove that the reaction at each joint is in a direction parallel to the opposite side of the triangle and of magnitude \[ \frac{W}{3\sqrt{6}}. \]