Problems

Prove that the segments cut off on any straight line by (1) a hyperbola, and (2) its asymptotes, have the same midpoint, and shew that the product of the two segments between the points of section by the asymptotes and one of the points on the curve is equal to the square of the parallel semi-diameter. Shew how to construct the hyperbola touching three given straight lines and having a given asymptote.

A conic is drawn touching two parallel straight lines at \(A\) and \(B\) respectively. Any third straight line is drawn cutting the conic in \(X\) and \(Y\), and the tangent at \(B\) in \(Z\). Shew that if the tangents at \(X\) and \(Y\) cut the tangent at \(A\) in \(L\) and \(M\) respectively, and if \(N\) be a point on the tangent at \(A\) such that \(NZ\) is also a tangent, then \(N\) bisects \(LM\).

The sides of an acute-angled triangle each subtend a right angle at some point not in the plane of the triangle. Shew that there are two possible positions of the point and that their join passes through the orthocentre of the triangle.

Determine the number of normals which can be drawn to an ellipse from a point in its plane, and establish the concurrency of normals to the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) at its points of intersection with any conic whose equation is of the form \[ \frac{x^2}{a^4}+\frac{y^2}{b^4}-1+axy+\beta x+\gamma y = 0, \] where \(\alpha, \beta, \gamma\) are arbitrary constants. Shew further that if the point of concurrency be \((\xi, \eta)\), then the axes of the two parabolas through the four corresponding points on the ellipse, intersect in the point \[ \left(\frac{a^2\xi}{a^4-b^4}, -\frac{b^4\eta}{a^4-b^4}\right). \]

Shew that if four points are chosen so that two rectangular hyperbolas can be drawn to pass through them, then every conic passing through them is also a rectangular hyperbola. Prove that for four concyclic points on a rectangular hyperbola, the centre of the hyperbola bisects the join of one of the points to the orthocentre of the other three, and that the centre of mean position of the four points bisects the join of the centres of the hyperbola and the circle through the four points.

Find the necessary and sufficient condition that \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] shall represent a pair of straight lines. Determine further conditions such that \[ a'x^2+2h'xy+b'y^2+2g'x+2f'y+c'=0 \] shall represent a pair of straight lines which with the first pair form a harmonic pencil. Hence or otherwise determine the equation of the bisectors of the angles between the first pair of lines.

(i) Shew that the radii of the circles touching the sides of a triangle are the roots of the equation \[ x^4 - x^3(4R+2r) + x^2(4Rr+r^2+s^2) - 2xs^2r+s^2r^2=0, \] where \(R\) is the circumradius, \(r\) the inradius, \(2s\) the perimeter of the triangle. (ii) If the centre of the escribed circle touching the side \(BC\) of a triangle \(ABC\) between \(B\) and \(C\) is joined to the vertices of the triangle, shew that the area of the triangle whose vertices are the three points in which these joins respectively meet the escribed circle is \[ \frac{1}{2}r_1^2\left\{\sin\frac{B}{2}+\sin\frac{C}{2}-\cos\frac{A}{2}\right\}, \] \(r_1\) being the radius of the escribed circle.

A balloon rises from level ground at a point whose bearing from a point \(A\) on the ground is \(20^\circ\) E. of N. The balloon travels in a direction \(\theta^\circ\) S. of W. with constant horizontal and vertical speeds, and when it is due North of \(A\) its vertical height is \(h\) and its elevation as seen from \(A\) is \(\alpha\). Shew that the elevation of the balloon when its bearing from \(A\) is \(\theta^\circ\) E. of N. is \(\tan^{-1}\left\{\frac{\sec\theta\tan\alpha}{2}\right\}\). Shew further that the shortest distance of the balloon from \(A\) is \[ h\cot\alpha\cos\theta\left\{\frac{1-\cos^2\alpha\cos^2 2\theta}{\cos^2 3\theta - \cos^2\alpha\cos\theta\cos 5\theta}\right\}^{\frac{1}{2}}. \]

Prove Demoivre's theorem for a rational index and shew how to express \(\cos\theta\) and \(\sin\theta\) in terms of exponential functions. Express \(\cos^n\theta\) (\(n\) being a positive integer) linearly in terms of cosines of multiples of \(\theta\), distinguishing if necessary between odd and even values of \(n\), and deduce the corresponding expressions for \(\sin^n\theta\).

A plane polygon of \(n\) sides of lengths \(a_1, a_2, \dots, a_n\), respectively, has angles given by \(\theta_{rs}\), the measure of the angle between the two sides \(a_r, a_s\), positively drawn in the same sense. Establish the relation \[ a_n^2=a_1^2+\dots+a_{n-1}^2+2\Sigma a_r a_s\cos\theta_{rs}, \] where in the summation all possible combinations of \(r\) and \(s\) are taken for \(r