Problems

Prove that the anharmonic ratio of the points in which a variable tangent cuts four fixed tangents to a conic is constant. Find the anharmonic ratio in which a variable tangent to the parabola \(y^2 = 4ax\) cuts the tangents at the points \((at_r^2, 2at_r)\), where \(r=1, 2, 3, 4\).

Find the condition that the line \(lx+my+n=0\) should be (i) a tangent, (ii) a normal to \(x^2/a^2+y^2/b^2=1\). The normal at the point on this ellipse whose eccentric angle is \(\phi\) meets the ellipse again in \(P\), prove that the tangent to the ellipse at \(P\) is \[ \frac{x}{a}\cos\phi\left(1 - \frac{c^2}{a^2}\cos^2\phi - \frac{c^2}{b^2}\sin^2\phi\right) + \frac{y}{b}\sin\phi\left(1 + \frac{c^2}{a^2}\cos^2\phi + \frac{c^2}{b^2}\sin^2\phi\right) + \frac{c^2}{ab}\left(\frac{\cos^2\phi}{a^2}+\frac{\sin^2\phi}{b^2}\right) = 0, \] where \(c^2=a^2-b^2\).

If \(lx+my+n=0\) is a straight line referred to rectangular axes, interpret the equations:

1. \(al+bm+cn=0\),
2. \((al+bm+cn)(a'l+b'm+c'n)=(a''l+b''m+c''n)^2\),
3. \((al+bm+cn)(a'l+b'm+c'n)=\kappa(l^2+m^2)\).

Find the coordinates of the centre of the conic in (iii).

Find the condition that the line \(lx+my+nz=0\) should touch the conic \[ ax^2+by^2+cz^2+2fyz+2gzx+2hxy=0, \] and, if the coordinates are areals, find the condition that the conic is a parabola. Prove that, the coordinates being areals, the locus of the pole of the line \[ lx+my+nz=0, \] with respect to the system of parabolas circumscribing the triangle of reference, is \[ \{x(my+nz-lx)\}^{1/2} + \{y(nz+lx-my)\}^{1/2} + \{z(lx+my-nz)\}^{1/2} = 0. \]

Find an expression for all angles which have the same sine as \(\alpha\). Find all the solutions of the equation \[ \sin 4\theta - \cos 3\theta = 0. \] Prove that \[ \sin^2 \frac{\pi}{14} + \sin^2 \frac{3\pi}{14} + \sin^2 \frac{5\pi}{14} = \frac{5}{4}. \]

Prove that

1. \(1-\cos^2\alpha - \cos^2\beta - \cos^2\gamma - 2\cos\alpha\cos\beta\cos\gamma\) \[ = -4\cos\frac{1}{2}(\alpha+\beta+\gamma)\cos\frac{1}{2}(-\alpha+\beta+\gamma)\cos\frac{1}{2}(\alpha-\beta+\gamma)\cos\frac{1}{2}(\alpha+\beta-\gamma). \]
2. \(\cos\alpha+\cos\beta+\cos\gamma+\cos(\alpha+\beta+\gamma) = 4\cos\frac{1}{2}(\beta+\gamma)\cos\frac{1}{2}(\gamma+\alpha)\cos\frac{1}{2}(\alpha+\beta)\).

Eliminate \(\theta\) from \begin{align*} a\sin 2\theta &= p\cos\theta+q\sin\theta, \\ b\cos 2\theta &= p\cos\theta-q\sin\theta. \end{align*}

In a triangle prove that, with the usual notation,

1. \(1/r_1 + 1/r_2 + 1/r_3 = 1/r\),
2. \(16R r r_1 r_2 r_3 = a^2b^2c^2\).

The intercepts made by the inscribed circle on the lines drawn from the vertices of a triangle \(ABC\) to the circum-centre are of lengths \(\alpha, \beta, \gamma\). Prove that \[ \frac{1}{\alpha^2} + \frac{1}{\beta^2} + \frac{1}{\gamma^2} = \frac{1}{4r^2} - \frac{1}{8rR \cos A \cos B \cos C}. \]

Two circles intersect in \(P\) and \(Q\). Draw a straight line through \(P\) so that the segments of the line cut off by the circles shall be in a given ratio.

A circle is inscribed in a right-angled triangle and another is escribed to one of the sides containing the right angle. Prove that the lines joining the points of contact with the hypotenuse and that side in each circle intersect one another at right angles, and being produced pass each through the point of contact of the other circle with the remaining side.

Prove that the inverse of a circle is a circle or a straight line, and find in each case the point into which the centre of the given circle inverts. A circle of radius \(a\) and centre \(C\) is inverted from a point \(O\) with respect to a circle of radius \(K\) into a circle of radius \(a'\) and centre \(C'\), and \(P'\) is the inverse of a point \(P\) in the same plane as the circles. Prove that \[ OP^2(a'^2-C'P'^2)(OC^2-a^2) = K^4 OP'^2(a^2-CP^2). \]