Problems

Prove geometrically that \(\tan\theta = \csc 2\theta - \cot 2\theta\). If \(\alpha+\beta+\gamma = \frac{\pi}{3}\), prove that \begin{align*} \cos(\beta+\gamma-2\alpha)\sin(\alpha+2\beta) + \cos(\gamma+\alpha-2\beta)\sin(\beta+2\gamma) + \cos(\alpha+\beta-2\gamma)\sin(\gamma+2\alpha) \\ = 4\sin(\alpha+2\beta)\sin(\beta+2\gamma)\sin(\gamma+2\alpha). \end{align*}

If P is the orthocentre of the triangle ABC, O its circumcentre and I its incentre, prove that

1. [(i)] \(IP^2 = 2r^2-4R^2\cos A\cos B\cos C\),
2. [(ii)] the area of the triangle POI \(= 2R^2\sin\frac{B-C}{2}\sin\frac{C-A}{2}\sin\frac{A-B}{2}\),

where R and r are the radii of the circumcircle and the incircle.

Prove that

1. [(i)] \(\cos\frac{\pi}{7}+\cos\frac{3\pi}{7}+\cos\frac{5\pi}{7}=\frac{1}{2}\),
2. [(ii)] \(\sec^2\frac{\pi}{7}+\sec^2\frac{3\pi}{7}+\sec^2\frac{5\pi}{7}=24\).

Find the exponential values of \(\cos\theta\) and \(\sin\theta\) and determine a general form for the values of \(\log_e(\cos\theta+i\sin\theta)\). Prove that if \((a+ib)^{x+iy}=q\), then \[ \frac{y}{x} = \frac{2\tan^{-1}\frac{b}{a}}{\log_e(a^2+b^2)}. \]

Determine the condition that the general equation of the second degree \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] should represent two straight lines, and find the coordinates of their point of intersection, when they are not parallel. Prove that, if the line joining their point of intersection to the origin bisects one of the angles between them, then \(h(f^2-g^2)=fg(b-a)\).

Prove that in general three normals can be drawn to a parabola through a given point. ABC is an equilateral triangle inscribed in a parabola of which A is the vertex. The normals at two points P and Q on the parabola meet in B. Prove that the length of PQ is twice the latus-rectum of the parabola and that the orthocentre of the triangle BPQ lies in AC.

Prove that the common chords of an ellipse and a circle, taken in pairs, are equally inclined to the axes of the ellipse. Prove that, if a circle touches the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) at the point P, whose eccentric angle is \(\phi\), and the equation of one common chord through P is \(\frac{x}{a}\cos\alpha+\frac{y}{b}\sin\alpha=\cos(\phi-\alpha)\), the equation of the other common chord through P is \(\frac{x}{a}\cos\alpha-\frac{y}{b}\sin\alpha=\cos(\phi+\alpha)\), and that the equation of the circle is \[ x^2+y^2-2(a^2-b^2)\left(\frac{x}{a}\cos\alpha\cos\phi - \frac{y}{b}\sin\alpha\sin\phi\right)+(a^2-b^2)\cos 2\alpha = a^2\sin^2\phi+b^2\cos^2\phi. \] Hence deduce the equation of the circle of curvature at the point whose eccentric angle is \(\phi\).

Prove that, in areal coordinates, the equation \[ \frac{x}{a}(\frac{y}{b}\cos A - \frac{z}{c}\cos B - \frac{x}{a}\cos C) = \frac{yz}{bc} \] is that of a circle, and find its radius and the coordinates of its centre. Find the chord joining the points of intersection of this circle and the circle \[ \frac{x^2}{a^2}\cos A + \frac{y^2}{b^2}\cos B + \frac{z^2}{c^2}\cos C = 0, \] and shew that the two circles cut one another orthogonally.

Explain the meanings of the partial differential coefficients \(\frac{\partial r}{\partial x}\) and \(\frac{\partial x}{\partial r}\) where \(x=r\cos\theta, y=r\sin\theta\), and give a geometrical illustration. Prove that \[ \frac{\partial^2 r}{\partial x^2}\frac{\partial^2 r}{\partial y^2} = \left(\frac{\partial^2 r}{\partial x\partial y}\right)^2. \]

If the coordinates of a point in a curve are known functions of a single parameter \(t\), find the equations of the tangent and normal at a given point in terms of the parameter. Prove that the equation of the normal to the curve \[ x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}} \] may be written in the form \(x\sin\phi-y\cos\phi+a\cos 2\phi=0\), and find the equation of the envelope of the normal.