Problems

If \(m<1\), and \(\theta\) and \(\phi\) are acute angles, and if \[ \theta = \phi - m\sin2\phi + \frac{1}{2}m^2\sin4\phi - \frac{1}{3}m^3\sin6\phi + \dots \text{ to infinity}, \] prove that \[ (1+m)\tan\theta = (1-m)\tan\phi. \]

If \(i=\sqrt{-1}\), if \(x, y, u\) and \(v\) are real quantities, and if \[ \tan(x+iy) = \sin(u+iv), \] prove that \[ \tan u \cdot \sinh 2y = \tanh v \cdot \sin 2x. \]

Two circles are given. Show how to construct a rhombus \(ABCD\) with \(A, C\) on one circle and \(B, D\) on the other. Show that all such rhombuses have equal sides.

\(OM, ON\) are fixed lines through \(O\), a point on a hyperbola. Through \(P\), a variable point on the hyperbola, a line \(PM\) is drawn parallel to one of the asymptotes to meet \(OM\) in \(M\), and \(PN\) is drawn parallel to the other asymptote to meet \(ON\) in \(N\). Prove that \(MN\) passes through a fixed point.

\(A, B\) are conjugate points with respect to a conic. \(R\) is a variable point on the conic and \(RA, RB\) meet the conic again in \(P, Q\). Show that \(PQ\) passes through a fixed point \(C\). Show that the triangles \(ABC, PQR\) are in perspective and that as \(R\) varies the centre of perspective describes a conic.

Prove that the polar reciprocal of a circle \(C\) with respect to another circle \(K\) is a conic \(C'\) whose focus is the centre of \(K\) and whose directrix is the polar with respect to \(K\) of the centre of \(C\).

Determine the length of the perpendicular let fall from any point \((h,k)\) on the line \(ax+by+c=0\). Prove that the product of the perpendiculars from \((h,k)\) on the \(n\) straight lines represented by the equation \[ a_0x^n + a_1x^{n-1}y + a_2x^{n-2}y^2 + \dots + a_ny^n = 0 \] is \[ \frac{a_0h^n + a_1h^{n-1}k + a_2h^{n-2}k^2 + \dots + a_nk^n}{\sqrt{(\{a_0-a_2+a_4-\dots\}^2 + \{a_1-a_3+a_5-\dots\}^2)}}. \]

Find the equation of the director circle of the conic \[ ax^2+2hxy+by^2+2gx+2fy+c=0, \] the axes being rectangular.

Prove that through any point two conics confocal with \(x^2/a^2+y^2/b^2=1\) can be drawn and express the coordinates of the point in terms of the semi-axes of the confocals. Show that the locus of a point such that the tangents from it to the ellipse contain an angle \(2\alpha\) is given by the equation \[ a_1^2 \cos^2\alpha + a_2^2 \sin^2\alpha = a^2, \] where \(a_1, a_2\) are the primary semi-axes of the confocals through the point.

Find the equations of the tangent and normal at any point of the conic \[ l/r = 1+e\cos\theta. \] A system of conics have a common focus and directrix. Show that the normals at the points where a line through the focus in a given direction meets the conics envelope a parabola, having its vertex at the focus and touching the given line.