Problems

If \(O, D, E\) and \(F\) are the centres of the inscribed and escribed circles of a triangle, prove that \(O\) is the orthocentre of the triangle \(DEF\). Prove also that every conic passing through the points \(O, D, E, F\) is a rectangular hyperbola.

Tangents from \(P\) to a given circle meet the tangent at a given point \(A\) in \(Q\) and \(R\). If the perpendicular distance from \(P\) to the tangent at \(A\) is constant, prove that the rectangle \(QA.AR\) is constant.

Any tangent to the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) meets the ellipse \(\frac{x^2}{a}+\frac{y^2}{b}=a+b\) in \(P\) and \(Q\). Prove that the tangents at \(P\) and \(Q\) are perpendicular.

A conic has a focus at the centre of a given circle; its eccentricity, and the direction of its major axis are also given. Tangents are drawn to it at the points where it meets the circle. For all such conics, prove that these tangents envelope a conic whose major axis is a diameter of the given circle. \par What happens if the given eccentricity is unity?

The tangents from \(P\) to the conic \(ax^2+by^2=1\) are harmonic conjugates with respect to the tangents from \(P\) to the conic \(ax^2-cy^2=-1\). Prove that the locus of \(P\) is two parallel straight lines.

The equation of a conic in homogeneous coordinates is \[ ax^2+by^2+cz^2+2fyz+2gzx+2hxy=0. \] Find the condition that the pole of \(lx+my+nz=0\) should lie on \(l'x+m'y+n'z=0\).

State and prove the rule for the multiplication of two determinants. \par Hence shew that the product \[ (x^3+y^3+z^3-3xyz)(a^3+b^3+c^3-3abc) \] may be expressed in the form \[ A^3+B^3+C^3-3ABC. \]

Solve the equation \[ 81x^4 + 54x^3 - 189x^2 - 66x + 40 = 0, \] given that the roots are in arithmetic progression.

Prove that, if \(m\) is a positive integer, \[ (\cos x+i\sin x)^m = \cos mx + i\sin mx. \] Sum the infinite series \[ 1+x^2\cos 2x + x^4\cos 4x + \dots \quad (x^2<1). \]

Eliminate \(\theta\) and \(\phi\) between the equations \begin{align*} a\sec\theta+b\cosec\theta &= c \\ a\sec\phi+b\cosec\phi &= c \\ \theta+\phi &= 2\alpha. \end{align*}