Find the equation of the director circle of the conic \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] the axes being rectangular. Shew that the director circles of the family of conics which touch the four lines \(x-a=0, x+a=0, bx-ay=0, cx-ay=0\), form the coaxal system through the two points given by \(x^2=a^2+bc, y=0\).
Prove that the triangle formed by the polars with regard to a conic of the vertices of another triangle is in perspective with that other triangle and that, if two triangles be in perspective, there is one conic with regard to which the sides of either triangle are the polars of the vertices of the other.
Prove that if \(a, b, c, \dots\) be any number of quantities, \(\Sigma a^3 - 3\Sigma abc\) is divisible by \(\Sigma a\), and find the quotient.
Shew that the \(n\)th convergent to the continued fraction \[ \frac{1}{1+} \frac{1}{2+} \frac{1}{1+} \frac{1}{2+\dots} \text{ is } 2 \frac{(1+\sqrt{3})^n - (1-\sqrt{3})^n}{(1+\sqrt{3})^{n+1} - (1-\sqrt{3})^{n+1}}. \]
\(A_1A_2\dots A_n\) is a regular polygon of \(n\) sides inscribed in a circle of radius \(a\). Prove that \[ A_1A_2^2 + A_1A_3^2 + \dots + A_1A_n^2 = 2na^2. \]
Find the \(n\)th differential coefficients with respect to \(x\) of \(\log(1+x^2)\) and \(e^x\sin^3x\).
Find the polar equation of the tangent and normal at any point of a given curve. If \(r, r'\) are the radii vectores to two fixed points, and \(\theta, \theta'\) the angles which they make with a fixed line, prove that the curves \(rr'=a\) and \(\theta+\theta'=b\) cut each other orthogonally.
Prove the formula for the radius of curvature at any point of a curve, using polar co-ordinates. Find the radius of curvature at any point of the curve \(r^2=a^2\cos 2\theta\) and prove that its evolute is \[ 9(x^{4/3}+y^{4/3})(x^{2/3}-y^{2/3})=4a^2. \]
If \((xy+tz)^2=x^3t^2(y+t)\), prove that \[ x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y} + t\frac{\partial z}{\partial t} = 2z + \frac{xy}{t}. \]
Two lines \(y=\pm mx\) meet the cubic \(x^3+y^3=3axy\) in points \(P, Q\) distinct from the origin. Prove that the tangents at \(P, Q\) meet on the curve.