The straight line \(x\cos\alpha+y\sin\alpha=p\) being called the line \((\alpha p)\), find the equation to the circle circumscribing the triangle formed by the lines \((\alpha p), (\beta q), (\gamma r)\), and shew that if it passes through the origin then \[ qr\sin(\beta-\gamma)+rp\sin(\gamma-\alpha)+pq\sin(\alpha-\beta) = 0. \]
Explain how to construct a circle (a) to pass through two given points and to touch a given straight line, (b) to pass through one given point and to touch two given straight lines, (c) to pass through one given point and to touch a given straight line at a given point of the line. Justify your constructions theoretically and examine any exceptional cases which may arise.
Prove that two homographic ranges are mutually projective. \(P, Q, R\) are three fixed collinear points and \(BC, CA, AB\) are three straight lines drawn through \(P, Q, R\) respectively. Prove that if \(B\) and \(C\) are constrained to move on two fixed intersecting lines then the locus of \(A\) is also a straight line. What result is obtained by reciprocating this theorem?
(i) \(p_1, p_2, p_3, p_4\) are the lengths of the perpendiculars from the vertices of a tetrahedron \(ABCD\) on to the opposite faces and \(\varpi_1, \varpi_2, \varpi_3, \varpi_4\) are the lengths of the perpendiculars from the centre of the circumscribed sphere on to these faces. Prove that \[ \frac{\varpi_1}{p_1}+\frac{\varpi_2}{p_2}+\frac{\varpi_3}{p_3}+\frac{\varpi_4}{p_4}=1. \] (ii) Given the lengths of the edges of a tetrahedron, indicate without obtaining any formula how the radius of the circumscribed sphere can be determined. \(X, Y, Z\) are three fixed points on a given sphere of radius \(R\). A fourth point \(P\) is taken on the sphere so that the volume of the tetrahedron \(PXYZ\) is a maximum. Prove that \(PX^2\) is the larger root of the equation \(4\Delta^2 x^2 - 16R^2\Delta^2 x + R^2 a^2 b^2 c^2 = 0\), where \(a, b, c, \Delta\) are respectively the lengths of the sides and the area of the triangle \(XYZ\).
Two straight lines passing through a given point \(P\) intersect a given ellipse in four concyclic points. Prove that the locus of the centre of the circle through these four points is the straight line through \(P\) perpendicular to the polar of \(P\) with respect to the ellipse.
In general, how many normals can be drawn from a given point to a rectangular hyperbola? Examine the case when the given point lies on the transverse axis of the hyperbola at a distance from the centre equal to twice the distance of the vertex from the centre. The normals at four points \(P, Q, R, S\) of a rectangular hyperbola all pass through the same point \(T\). The circumcircle of the triangle \(QRS\) intersects the hyperbola again in \(U\). Prove that \(PU\) passes through the centre of the hyperbola. If \(PU\) also passes through \(T\), what is the locus of \(T\)?
The equations of two intersecting straight lines are \[ a_1x+b_1y+c_1=0 \quad \text{and} \quad a_2x+b_2y+c_2=0, \] where \(c_1\) and \(c_2\) are both positive. Find the equation of the line bisecting the angle between these lines which contains the origin. Show further that the sign of \(a_1a_2+b_1b_2\) determines whether this is also the bisector of the acute angle between the lines. Find the coordinates of the incentre of the triangle formed by the lines whose equations are \[ y=2x-3, \quad 2y=x+2, \quad 2x+3y=24. \]
A family of ellipses have a common minor axis. Prove that the polars of a given point \(P\) with respect to the ellipses are concurrent. If \(Q\) is the point of concurrence and if \(PQ\) is equally inclined to the axes of the ellipses, prove that the locus of \(P\) consists of two hyperbolas and find their eccentricities.
\(A, B, C\) are three given non-collinear points. Prove that three circles can be drawn with \(A, B, C\) as centres and such that each circle touches the other two. Prove that the incircle of the triangle \(ABC\) intersects each of these three circles orthogonally. If the angles of the triangle \(ABC\) are in arithmetical progression with small common difference \(k\), prove that the ratio of the total area of the sectors of the circles included inside the triangle \(ABC\) to the area of the triangle \(ABC\) is \[ \frac{\pi}{2\sqrt{3}} + \frac{7\pi k^2}{9\sqrt{3}} - \frac{4k^2}{3}, \] neglecting powers of \(k\) higher than the second.
Define the hyperbolic functions and establish their most important properties, including the expressions for \(\sinh(u+v)\) and \(\cosh(u+v)\) in terms of \(\sinh u, \sinh v, \cosh u, \cosh v\). Prove that the sum of the series \[ \sinh\theta+\tan\theta\sinh 2\theta+\tan^2\theta\sinh 3\theta+\dots+\tan^{n-1}\theta\sinh n\theta \] is \[ \frac{\sinh\theta-\tan^n\theta\sinh(n+1)\theta+\tan^{n+1}\theta\sinh n\theta}{\sec^2\theta-2\cosh\theta\tan\theta}. \]