\(A\), \(B\), \(C\), \(D\), \(E\) and \(P\) are six points in general position in a plane. Describe and justify straight line constructions for finding the other intersections \(A_1\) and \(B_1\) of the lines \(PA\), \(PB\) with the conic \(S\) through \(A\), \(B\), \(C\), \(D\), \(E\), and deduce a construction for the polar line of \(P\) with respect to \(S\). Give also without proof a construction for the tangent to \(S\) at \(A\).
From a point \(O\) perpendiculars \(OA'\), \(OB'\), \(OC'\), \(OD'\) are drawn to the faces of a tetrahedron \(ABCD\). Prove that pairs of lines such as \((AB, C'D')\), \((BC, A'D')\) are mutually perpendicular. Hence prove that any pair of perpendiculars from \(A\), \(B\), \(C\), \(D\) to the corresponding faces of the tetrahedron \(A'B'C'D'\) are coplanar, and deduce that all these perpendiculars are concurrent.
A circle of radius \(r\) rolls completely round the outside of a closed convex curve \(\mathscr{C}\) of length \(2\pi r\). Show (i) that the centre of the circle traces out a curve \(\mathscr{D}\) of length \(4\pi r\), and (ii) that the region inside \(\mathscr{D}\) but outside \(\mathscr{C}\) has area \(3\pi r^2\).
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.