Express \(\sqrt{12}\) as a simple continued fraction, and shew that, if \(u, u'\) are successive convergents, \(\displaystyle\frac{u'}{3} = 1 + \frac{1}{u+3}\).
Define a system of coaxal circles. Prove that one circle of the system can be drawn through any given point, also that the polars of a given point with respect to the circles of the system are concurrent.
Three straight lines meet in a point but are not in the same plane. Shew how to draw a straight line through the point making equal angles with the three.
Find the equation of the circle which has for a diameter the chord \(x=c\) of the hyperbola \(x^2+2mxy-y^2=a^2\). Hence shew that the circles described on parallel chords of a rectangular hyperbola as diameters are coaxal.
Find the condition that the general equation of the second degree should represent a parabola. Prove that the equation \((x+y\tan\alpha)^2=4ay\sec^3\alpha\) represents a parabola whose latus rectum is \(4a\).
Find the condition that the line \(y-y'=m(x-x')\) should touch the ellipse \(x^2/a^2+y^2/b^2=1\). Prove that it is normal to the ellipse if \[ (a^2+b^2m^2)(y'-mx')^2=(a^2-b^2)^2m^2. \]
Find the trilinear equation of the straight line drawn through the angular point \(A\) of the fundamental triangle perpendicular to the line \(l\alpha+m\beta+n\gamma=0\). Shew that the locus of the intersection of perpendicular lines through \(B\) and \(C\) respectively is \[ \alpha^2\cos A - \alpha(\beta\cos B+\gamma\cos C)-\beta\gamma=0. \]
Shew that the poles of a fixed straight line with reference to a system of confocal conics are collinear.
If \begin{align*} x^2 - yz + (a-\lambda)x &= 0, \\ y^2 - zx + (b-\lambda)y &= 0, \\ z^2 - xy + (c-\lambda)z &= 0, \end{align*} and \[ x^2y^2+y^2z^2+z^2x^2 = xyz(x+y+z), \] prove that \[ 3\lambda = a+b+c, \] and that \[ a^2+b^2+c^2 = bc+ca+ab. \]
State and prove the Binomial Theorem for a positive integral exponent. If \[ (1+x)^{4m} = 1+c_1x+c_2x^2+\dots, \] where \(m\) is a positive integer, prove that \[ 1+c_1+c_2+c_3+\dots = 2^{4m} \] and that \[ 1-c_2+c_4-c_6+\dots = (-1)^m \cdot 2^{2m}. \]