Two equal circles touch each other externally at a point \(O\), and the tangent at a general point \(P\) of one of them meets the other in \(Q\) and \(R\). Prove that \(OP^2 = OQ \cdot OR\).
(i) Show that in rectangular cartesian coordinates the equation $$p(x^4 + y^4) + qxy(x^2 - y^2) + rx^2y^2 = 0$$ represents always two pairs of straight lines at right angles. Find the condition that the two pairs will coincide. (ii) Find the area of the triangle formed by the lines whose equations in rectangular cartesian coordinates are $$ax^2 + 2hxy + by^2 = 0$$ and $$lx + my + 1 = 0.$$
Show that there exists a unique circle, \emph{the polar circle}, with respect to which a given triangle is self-polar. Determine its centre and radius, and state the conditions in which it is a real or imaginary circle. Prove that this polar circle is coaxial with the circumcircle and nine-points circle of the triangle.
Obtain necessary and sufficient conditions that two circles in different planes shall be sections of the same sphere. One of two coplanar circles is rotated about their radical axis and brought into a different plane, the other circle meanwhile remaining fixed. Show that in the changed position the circles are sections of a common sphere.
Find the equation of the tangent at the point \(\theta = \alpha\) in polar coordinates to the conic of equation \(l/r = 1 + e\cos\theta\). Two fixed conics have common focus \(S\), and their axes are inclined at an angle \(\beta\). Two points \(P\) and \(Q\) are taken, one on each conic, such that \(PSQ = 90^\circ\), and the tangents at \(P\) and \(Q\) meet in \(T\). Show that the locus of \(T\) is a conic. If \(e\) and \(e'\) are the eccentricities of the given conics, find the condition that the locus of \(T\) is a parabola.
Show that all real conics concentric with the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) and orthogonally belong to one or other of the families $$\frac{x^2}{a^2 + \lambda} + \frac{y^2}{b^2 + \lambda} = 1 \quad \text{and} \quad \frac{x^2 + 2lcxy - y^2}{l^2} = \frac{a^2 - l^2}{a^2 + b^2}.$$
If a conic is inscribed in the triangle of reference of areal coordinates, show that its equation can be reduced to the form $$\sqrt{lx} + \sqrt{my} + \sqrt{nz} = 0.$$ Find the coordinates of its centre.
Show that the curve in rectangular coordinates of parametric equations $$x = at^2 + 2bt + c, \quad y = a't^2 + 2b't + c',$$ is a parabola of latus rectum \(2(a'b - ab')/(a^2 + a'^2)^{1/2}\). Find also the equation of the directrix of the parabola.
If for a triangle \(ABC\) the circumcentre is \(O\) and the orthocentre is \(H\), show that $$OH^2 = R^2(1 - 8\cos A \cos B \cos C),$$ where \(R\) is the radius of the circumcircle. Hence show that the circumcircle and nine-points circle of a triangle intersect in distinct real points only if the triangle is obtuse, and find an expression for the angle at which they then intersect.
Prove that \(\tan^2(\pi/11)\) is a root of the equation $$x^5 - 55x^4 + 330x^3 - 462x^2 + 165x - 11 = 0,$$ and state what are its other roots. By expressing the left-hand side in terms of \(\tan(\pi/11)\), or otherwise, prove that $$\tan(3\pi/11) + 4\sin(2\pi/11) = \sqrt{11}.$$