Find the necessary and sufficient condition that \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] shall represent a pair of straight lines. Determine further conditions such that \[ a'x^2+2h'xy+b'y^2+2g'x+2f'y+c'=0 \] shall represent a pair of straight lines which with the first pair form a harmonic pencil. Hence or otherwise determine the equation of the bisectors of the angles between the first pair of lines.
(i) Shew that the radii of the circles touching the sides of a triangle are the roots of the equation \[ x^4 - x^3(4R+2r) + x^2(4Rr+r^2+s^2) - 2xs^2r+s^2r^2=0, \] where \(R\) is the circumradius, \(r\) the inradius, \(2s\) the perimeter of the triangle. (ii) If the centre of the escribed circle touching the side \(BC\) of a triangle \(ABC\) between \(B\) and \(C\) is joined to the vertices of the triangle, shew that the area of the triangle whose vertices are the three points in which these joins respectively meet the escribed circle is \[ \frac{1}{2}r_1^2\left\{\sin\frac{B}{2}+\sin\frac{C}{2}-\cos\frac{A}{2}\right\}, \] \(r_1\) being the radius of the escribed circle.
A balloon rises from level ground at a point whose bearing from a point \(A\) on the ground is \(20^\circ\) E. of N. The balloon travels in a direction \(\theta^\circ\) S. of W. with constant horizontal and vertical speeds, and when it is due North of \(A\) its vertical height is \(h\) and its elevation as seen from \(A\) is \(\alpha\). Shew that the elevation of the balloon when its bearing from \(A\) is \(\theta^\circ\) E. of N. is \(\tan^{-1}\left\{\frac{\sec\theta\tan\alpha}{2}\right\}\). Shew further that the shortest distance of the balloon from \(A\) is \[ h\cot\alpha\cos\theta\left\{\frac{1-\cos^2\alpha\cos^2 2\theta}{\cos^2 3\theta - \cos^2\alpha\cos\theta\cos 5\theta}\right\}^{\frac{1}{2}}. \]
Prove Demoivre's theorem for a rational index and shew how to express \(\cos\theta\) and \(\sin\theta\) in terms of exponential functions. Express \(\cos^n\theta\) (\(n\) being a positive integer) linearly in terms of cosines of multiples of \(\theta\), distinguishing if necessary between odd and even values of \(n\), and deduce the corresponding expressions for \(\sin^n\theta\).
A plane polygon of \(n\) sides of lengths \(a_1, a_2, \dots, a_n\), respectively, has angles given by \(\theta_{rs}\), the measure of the angle between the two sides \(a_r, a_s\), positively drawn in the same sense. Establish the relation
\[ a_n^2=a_1^2+\dots+a_{n-1}^2+2\Sigma a_r a_s\cos\theta_{rs}, \]
where in the summation all possible combinations of \(r\) and \(s\) are taken for \(r
If \(P, Q,\) and \(R\) are points on the sides \(BC, CA,\) and \(AB\) respectively of a triangle \(ABC\) such that \[ \mu.BP+\nu.CP=\nu.CQ+\lambda.AQ=\lambda.AR+\mu.BR=0, \] shew that the lines \(AP, BQ\) and \(CR\) meet in a point \(O\) and that \[ \lambda.Al + \mu.Bl + \nu.Cl = (\lambda+\mu+\nu).Ol, \] where \(l\) is any line of the plane and \(Al\), for instance, denotes the perpendicular distance of \(A\) from \(l\), regard being had to sign. Hence shew that, if \(I, I_1, I_2\) and \(I_3\) are the centres of the inscribed and escribed circles of the triangle, \[ (s-a).I_1l + (s-b).I_2l + (s-c).I_3l = s.Il. \]
Shew that if \(A, B, C\) and \(A', B', C'\) are sets of points on two coplanar lines, then the points of intersection \((BC', B'C), (CA', C'A),\) and \((AB', A'B)\) are collinear. Three lines \(a, b,\) and \(c\) meet in a point \(I\), and \(d\) is any line not passing through \(I\). Two ranges of points \((A_1, A_2, \dots)\) and \((B_1, B_2, \dots)\) on \(a\) and \(b\) respectively are in perspective from a point \(O\). Shew that two points \(H\) and \(K\) can be found on \(c\) such that the ranges \((A_1, A_2, \dots)\) and \((B_1, B_2, \dots)\) are in perspective with the same range on \(d\), the centres of perspective being \(H\) and \(K\) respectively.
Define an involution of points on a line and shew that an involution is determined by two pairs of points. Chords joining two fixed points \(A\) and \(B\) of a conic to a variable point \(P\) of the conic meet any line \(l\) in pairs of points. Shew that these form an involution only if \(l\) goes through the pole of \(AB\). Shew that if four coplanar circles meet in a point \(O\) the six lines which join \(O\) to the other intersections of the circles are three pairs of lines of an involution pencil.
Shew that the polar reciprocal of a conic with respect to a circle whose centre is at a focus is a circle. Shew that the reciprocals of a system of confocal conics with respect to a circle whose centre is one of the common foci are circles of a coaxal system. How do the limiting points arise, and what happens to the property that two confocal conics cut at right angles?
Shew that there are two spheres which touch a given right circular cone along circles and also touch a given plane \(\alpha\). The points of contact of the spheres with \(\alpha\) are \(S\) and \(S'\). The line \(SS'\) meets the cone in \(A\) and \(A'\) and the planes of the circles of contact in \(K\) and \(K'\). Shew that \(A\) and \(A'\) are harmonically separated by \(S\) and \(K\) and also by \(S'\) and \(K'\) and that if \(P\) is any point on the section of the cone by the plane \(\alpha\), then either \[ SP+S'P = AA', \] or \[ SP \sim S'P = AA'. \]