If \(\alpha+\beta+\gamma=2m\pi\), where \(m\) is an integer, prove that \[ \cos^2\alpha+\cos^2\beta+\cos^2\gamma-2\cos\alpha\cos\beta\cos\gamma=1. \] If \(p,q,r\) are the respective distances of the vertices \(A, B, C\) of a triangle from a point \(O\) in the plane of \(ABC\), shew that \[ a^2b^2c^2 - \Sigma a^2p^2(b^2+c^2-a^2) + \Sigma a^2(p^2-q^2)(p^2-r^2)=0, \] where \(a,b,c\) are the sides of the triangle.
Prove that \[ \int_0^\infty \frac{dx}{x^2+2x\cos\alpha+1} = \frac{\alpha}{\sin\alpha} \qquad 0 < \alpha < \pi. \] Evaluate \(\displaystyle\int_0^\infty \frac{dx}{x^4+2x^2\cos\alpha+1}\) and \(\displaystyle\int_0^\infty \frac{(x^2+1)dx}{x^4+2x^2\cos\alpha+1}\).
If \(x\) and \(y\) are functions of \(\xi\) and \(\eta\), and \begin{align*} a &= \left(\frac{\partial x}{\partial\xi}\right)^2 + \left(\frac{\partial y}{\partial\xi}\right)^2, & b &= \left(\frac{\partial x}{\partial\eta}\right)^2 + \left(\frac{\partial y}{\partial\eta}\right)^2, & h &= \frac{\partial x}{\partial\xi}\frac{\partial x}{\partial\eta} + \frac{\partial y}{\partial\xi}\frac{\partial y}{\partial\eta}, \end{align*} and if \[ H = \frac{\partial x}{\partial\xi}\frac{\partial y}{\partial\eta} - \frac{\partial x}{\partial\eta}\frac{\partial y}{\partial\xi}, \] shew that \[ H\frac{\partial U}{\partial x} = \frac{\partial U}{\partial\xi}\frac{\partial y}{\partial\eta} - \frac{\partial U}{\partial\eta}\frac{\partial y}{\partial\xi}, \quad H\frac{\partial U}{\partial y} = -\frac{\partial U}{\partial\xi}\frac{\partial x}{\partial\eta} + \frac{\partial U}{\partial\eta}\frac{\partial x}{\partial\xi}, \] where \(U\) is any differentiable function of \((x,y)\). Prove also that \[ \frac{\partial^2 U}{\partial x^2} + \frac{\partial^2 U}{\partial y^2} = \frac{1}{H}\frac{\partial}{\partial\xi}\left\{\frac{b\dfrac{\partial U}{\partial\xi} - h\dfrac{\partial U}{\partial\eta}}{H}\right\} + \frac{1}{H}\frac{\partial}{\partial\eta}\left\{\frac{-h\dfrac{\partial U}{\partial\xi} + a\dfrac{\partial U}{\partial\eta}}{H}\right\}. \]
Polynomials \(f_0(x), f_1(x), f_2(x), \dots\) are defined by the relation \[ f_n(x) = \frac{d^n}{dx^n}(x^2-1)^n. \] Prove that \[ \int_{-1}^{+1} f_n(x) f_m(x) dx = 0 \] if \(m \neq n\), and that \[ \int_{-1}^{+1} \{f_n(x)\}^2 dx = \frac{2(n!)^2 2^{2n+1}}{2n+1}. \] Shew that if \(\phi(x)\) is any polynomial of degree \(m\), \[ \phi(x) = \sum_{n=0}^m a_n f_n(x), \] where \[ a_n = \frac{2n+1}{(n!)^2 2^{2n+1}} \int_{-1}^{+1} \phi(x) f_n(x) dx. \]
If \(b_1, b_2, b_3, \dots, b_n\) are numbers such that \[ b_1 \ge b_2 \ge b_3 \ge \dots \ge b_n, \] and \(a_1, a_2, a_3, \dots, a_n\) are such that \[ a_1+a_2+\dots+a_r \le S \quad (r=1,\dots,n) \] shew that \[ a_1b_1+a_2b_2+\dots+a_nb_n \le Sb_1. \] If \(b_1, b_2, b_3, \dots\) is a decreasing sequence such that \[ \lim_{n\to\infty} b_n = 0, \] shew that \[ \sum_1^\infty b_n\cos nx \] converges if \(x \neq 2m\pi\), where \(m\) is an integer.
If \(Y\) is the foot of the perpendicular from a focus of a central conic on to the tangent to the conic at any point and if \(H\) is the image of the focus in the tangent, prove that the loci of \(Y\) and \(H\) are circles. Obtain the corresponding results in the case of a parabola.
Prove that the inverse of a sphere with respect to any internal point is a sphere. Invert with respect to the orthocentre the theorem that the three perpendiculars of a triangle are concurrent.
Shew that the cross-ratio of the pencil \(u+\lambda_r v=0\), (\(r=1,2,3,4\)), is \[ \frac{(\lambda_1-\lambda_2)(\lambda_3-\lambda_4)}{(\lambda_1-\lambda_4)(\lambda_3-\lambda_2)}. \] The equations of the sides \(AB, AC\) of a triangle \(ABC\) are \(u=0\) and \(v=0\) respectively, and the equation of the internal bisector of the angle \(A\) is \(u+v=0\). Shew that the equation of the straight line through \(A\) and perpendicular to \(BC\) is \(u\cos C+v\cos B=0\). Obtain an expression for the cross-ratio of the pencil formed by these four lines.
\(2a\) and \(2b\) are respectively the lengths of the major and minor axes of an ellipse which touches a parabola of latus rectum \(4c\). When produced the major axis of the ellipse coincides with the axis of the parabola. Shew that the length intercepted by the points of contact on a common tangent to the two curves (other than the tangent at their point of contact) is \[ \frac{(2ac+b^2)^{\frac{1}{2}}(2ac+b^2+c^2)^{\frac{1}{2}}}{c(ac+b^2)^{\frac{1}{2}}}. \]
The straight line passing through the points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) intersects the ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\) in the points \(P\) and \(Q\). Obtain an equation whose roots are the ratios in which \(AB\) is divided by \(P\) and \(Q\). Hence find the equation of the pair of tangents drawn from \(A\) to the ellipse and deduce the equation of the director circle. Shew that the envelope of the chords of contact of tangents drawn to an ellipse of eccentricity \(e\) from points on its director circle is an ellipse whose eccentricity is \(e(2-e^2)^{-\frac{1}{2}}\).