Show that the polar equation of a conic referred to a focus as origin may be put in the form \[ l/r=1+e\cos\theta. \] Show also that the eccentric angle \(\phi\) of a point \(P\) of an ellipse is related to the radius vector from this focus by the equation \[ r = a(1-e\cos\phi). \] Hence, or otherwise, prove for the ellipse, that \[ \tan\frac{\theta}{2} = \sqrt{\frac{1+e}{1-e}}\tan\frac{\phi}{2}. \]
Prove that if the two triangles \(ABC, PQR\) both circumscribe a conic \(\Sigma\), their vertices all lie on another conic \(S\). Prove further that if any tangent to \(\Sigma\) cut \(S\) in points \(X\) and \(Y\), the other tangents to \(\Sigma\) from \(X\) and \(Y\) meet in a point on \(S\).
Prove that if \[ \sec\alpha = \sec\beta\sec\gamma + \tan\beta\tan\gamma, \] then either \[ \begin{cases} \sec\beta = \sec\gamma\sec\alpha + \tan\gamma\tan\alpha, \\ \sec\gamma = \sec\alpha\sec\beta + \tan\alpha\tan\beta; \end{cases} \] or \[ \begin{cases} \sec\beta = \sec\gamma\sec\alpha - \tan\gamma\tan\alpha, \\ \sec\gamma = \sec\alpha\sec\beta - \tan\alpha\tan\beta. \end{cases} \]
If the triangle \(ABC\) has sides of length \(a,b\), and \(c\), respectively, and if with the usual notation \(r, r_1, r_2\) and \(r_3\) denote the radii of the inscribed and escribed circles respectively, prove that \[ r_2 r_3 \tan\frac{1}{2}A \quad \text{and} \quad \frac{r_2-r_3}{b-c}, \] are equal, and determine their common value in terms of \(a,b\), and \(c\).
Find the limit, as \(x\) tends to zero, of \[ \frac{x\cos x - \sin x}{x^3}. \] Sketch the curve \[ y = \frac{\sin x}{x} \] and discuss briefly its form for small values of \(x\). Show that, for large values of \(x\), there are maxima and minima near the points \(x=(n+\frac{1}{2})\pi\), where \(n\) is an integer. Obtain a closer approximation to the values of \(x\) at the maxima and minima, with an error of order \(\dfrac{1}{n^3}\).
Show that the equation \[ x^4 + 3x^2 - 3 = 0 \] has one positive root. Find to three decimal places an approximation to this root.
A man standing on the edge of a circular pond wishes to reach the diametrically opposite point by running or swimming or a combination of both. Assuming that he can run \(k\) times as fast as he can swim, find what is his quickest method, for any given \(k>1\).
From the equations \(y=f(x)\), \(x=\xi\cos\alpha - \eta\sin\alpha\) and \(y=\xi\sin\alpha+\eta\cos\alpha\) (where \(\alpha\) is constant) it is deduced that \(\eta=\phi(\xi)\). Prove that \[ \frac{\frac{d^2y}{dx^2}}{\left[1+\left(\frac{dy}{dx}\right)^2\right]^{\frac{3}{2}}} = \frac{\frac{d^2\eta}{d\xi^2}}{\left[1+\left(\frac{d\eta}{d\xi}\right)^2\right]^{\frac{3}{2}}}, \] and interpret this result geometrically.
If \(y\) is defined as a function of \(x\) by the equation \(y\sqrt{1+x^2}=\log[x+\sqrt{1+x^2}]\), prove that \[ (1+x^2)y' + xy = 1 \] and express \(y\) as a series in ascending powers of \(x\). Hence show that the sum of the series \[ 1 - \frac{1}{3!} + \frac{(2!)^2}{5!} - \frac{(3!)^2}{7!} + \dots \] is \(\dfrac{4}{\sqrt{5}}\log\tfrac{1}{2}(1+\sqrt{5})\).
Evaluate the integrals