Explain what is meant by a projective correspondence (or homography) between the points on a straight line, and show that there is a unique projective correspondence in which to given points \(A, B, C\) there correspond given points \(A', B', C'\). \(M\) is a self-corresponding point of a projective correspondence on a line and to \(A, B\) there correspond respectively \(A', B'\). On any line through \(M\) two points \(S, S'\) are taken; \(SA, S'A'\) meet in \(P\) and \(SB, S'B'\) in \(Q\). Prove that the point \(N\) in which \(PQ\) meets the given line is the other self-corresponding point. Deduce, or prove otherwise, that if in a projective correspondence the two self-corresponding points coincide in \(M\) (a parabolic correspondence), and to \(A\) corresponds \(A'\) and to \(A'\) corresponds \(A''\), then \(M, A'\) are harmonically separated by \(A, A''\). (Assume the harmonic property of the quadrilateral.)
(i) Find the angle between the straight lines given by the equation (in rectangular Cartesian coordinates) \[ ax^2+2hxy+by^2=0. \] (ii) Show that the equation of the base of the triangle of which these lines are the two sides and whose orthocentre is the point \((x_0, y_0)\) is \[ (a+b)(xx_0+yy_0) = bx_0^2-2hx_0y_0+ay_0^2. \]
Prove that the straight line \[ ty = x+at^2 \] touches the parabola \(y^2=4ax\), and find the coordinates of the point of contact. Prove that the locus of the point of intersection of tangents to the parabola which intercept a fixed length \(l\) on the directrix is \[ (x+a)^2(y^2-4ax)=l^2x^2. \]
Find the condition that the straight lines \(l_1x+m_1y=1\), \(l_2x+m_2y=1\) should be conjugate (i.e. each pass through the pole of the other) with regard to the ellipse \[ x^2/a^2+y^2/b^2=1. \] Two points \(P, Q\) of the ellipse \(x^2/\alpha^2+y^2/\beta^2=1\) are such that the tangents at \(P\) and \(Q\) are conjugate with regard to the ellipse \(x^2/a^2+y^2/b^2=1\). Prove that the chord \(PQ\) touches the ellipse \[ \frac{x^2}{\alpha^4(b^2+\beta^2)} + \frac{y^2}{\beta^4(a^2+\alpha^2)} = \frac{1}{a^2\beta^2+b^2\alpha^2}. \]
The lengths of the semi-axes of an ellipse are \(\alpha, \beta\) (\(\alpha > \beta\)), its centre is at the origin of coordinates and its major axis makes an angle \(\phi\) with the \(x\)-axis. Find its equation. Prove that if two concentric ellipses touch one another the angle \(\theta\) between their major axes is given by \[ \tan^2\theta = \frac{(a^2-a^2)(b^2-\beta^2)}{(a^2-\beta^2)(\alpha^2-b^2)}, \] where \(a, b\) and \(\alpha, \beta\) are their semi-axes.
Find the condition that the straight line joining the two points \(P, Q\), whose homogeneous coordinates are \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\), should meet the conic \(\alpha x^2+\beta y^2+\gamma z^2=0\) in two points which are harmonically separated by \(P, Q\). A line \(lx+my+nz=0\) is such that it meets the two conics \(ax^2+by^2+cz^2=0\), \(\alpha x^2+\beta y^2+\gamma z^2=0\) in two pairs of points which are harmonically separated. Prove that \[ l^2(b\gamma+c\beta) + m^2(c\alpha+a\gamma) + n^2(a\beta+b\alpha) = 0. \]
Prove that if two coplanar triangles are such that the lines joining corresponding vertices are concurrent, then the points of intersection of corresponding sides are collinear. \(A, B, C\), and \(D\) are the vertices of a quadrangle. \(AB\) and \(CD\) meet at \(F\); \(AC, BD\) at \(G\); \(AD, BC\) at \(H\). \(CD\) meets \(GH\) at \(P\), \(DB\) meets \(HF\) at \(Q\), \(BC\) meets \(FG\) at \(R\). Prove that \(P, Q\) and \(R\) are collinear. Shew that there are four such lines, forming a quadrilateral, whose diagonal triangle is \(FGH\).
Prove that the lines \(\alpha=0, \alpha-\lambda\beta=0, \beta=0, \alpha+\lambda\beta=0\), where \(\lambda\) is a constant, form an harmonic pencil. If two conics, \(S\) and \(S'\), each have double contact with a third conic, prove that the chords of contact meet at the point of intersection of a pair of common chords of \(S\) and \(S'\), the four lines forming an harmonic pencil. Conversely, if \(\alpha=0\) and \(\beta=0\) are a pair of common chords of \(S\) and \(S'\), shew that there is a conic having double contact with \(S\) along \(\alpha-\lambda\beta=0\), and with \(S'\) along \(\alpha+\lambda\beta=0\).
The function \(y=\sin x\) satisfies the differential equation \(\frac{d^2y}{dx^2}+y=0\). Assuming that \(\sin x\) can be expanded as a series in ascending powers of \(x\), deduce the series. Prove that, if \(y=\sin(n\sin^{-1}x)\), then \[ (1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} + n^2y = 0. \] Deduce the expansion of \(y\) as a series in ascending powers of \(x\), when such an expansion is possible. Discuss what happens when \(n\) is an integer.
Prove by induction or otherwise that \begin{align*} \cos(\alpha_1+\alpha_2+\alpha_3+\dots+\alpha_n) &= \{1-s_2+s_4-s_6+\dots\}\cos\alpha_1\cos\alpha_2\dots\cos\alpha_n, \\ \sin(\alpha_1+\alpha_2+\alpha_3+\dots+\alpha_n) &= \{s_1-s_3+s_5-s_7+\dots\}\cos\alpha_1\cos\alpha_2\dots\cos\alpha_n, \end{align*} where \(s_r\) denotes the sum of the products of \(\tan\alpha_1, \tan\alpha_2, \dots \tan\alpha_n\) taken \(r\) at a time. If \(\theta_1, \theta_2, \theta_3\) and \(\theta_4\) are the four roots of \(a\cos2\theta+b\sin2\theta+c\cos\theta+d\sin\theta+e=0\) between 0 and \(2\pi\), prove that \[ \frac{\cos\frac{1}{2}(\theta_1+\theta_2+\theta_3+\theta_4)}{a} = \frac{\sin\frac{1}{2}(\theta_1+\theta_2+\theta_3+\theta_4)}{b} = \frac{\sum \cos\frac{1}{2}(-\theta_1+\theta_2+\theta_3+\theta_4)}{-c} = \frac{\sum\sin\frac{1}{2}(-\theta_1+\theta_2+\theta_3+\theta_4)}{-d}. \]