Simplify the expression \[ \frac{\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{(x+y)^2}{(a+b)^2}}{\frac{x}{a}+\frac{y}{b}-\frac{x+y}{a+b}}. \]
Prove that, if \(n_r\) is the number of combinations of \(n\) things taken \(r\) at a time, \[ \begin{vmatrix} n_r & n_{r+1} & n_{r+2} \\ (n+1)_r & (n+1)_{r+1} & (n+1)_{r+2} \\ (n+2)_r & (n+2)_{r+1} & (n+2)_{r+2} \end{vmatrix} = \frac{n_r(n+1)_r(n+2)_r}{(r+1)_r(r+2)_2}. \]
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.