Problems

Prove by reciprocation or otherwise that chords of a rectangular hyperbola that subtend a right angle at a focus \(S\) envelope a parabola whose focus is \(S\).

If every edge of a tetrahedron is perpendicular to the edge that it does not meet, prove that the perpendicular from any vertex on the opposite face passes through the orthocentre of that face.

Prove that the two straight lines \[ (x^2+y^2)(\cos^2\theta \cdot \sin^2\alpha + \sin^2\theta) = (x\tan\alpha - y\sin\theta)^2 \] meet at an angle \(2\alpha\).

Prove that the equation of the chord of the parabola \(y^2=4ax\) that has its middle point at \((x', y')\) is \[ yy' - 2ax = y'^2 - 2ax'. \] Any tangent to \(y^2+4bx=0\) meets \(y^2=4ax\) at \(P\) and \(Q\). Prove that the locus of the middle point of \(PQ\) is \[ y^2(2a+b) = 4a^2x. \]

Prove that the product of the perpendiculars drawn to the normal at a point \(P\) of an ellipse from the centre and from the pole of the normal is equal to the product of the focal distances of \(P\).

In homogeneous coordinates, find an equation for the system of conics that touches the four straight lines \[ px \pm qy \pm rz = 0. \] Find also the locus of the poles with respect to these conics of the line \[ lx+my+nz=0. \]

Prove that the equation \[ ax^2+2hxy-by^2 = 0 \] represents a pair of conjugate diameters of the ellipse \[ ax^2+by^2=1. \] Prove also that there is one conic with these lines as asymptotes that cuts \(ax^2+by^2=1\) orthogonally at their four points of intersection, and find its equation.

Express in partial fractions \[ \frac{(x-\alpha)(x-\beta)(x-\gamma)(x-\delta)}{(x-a)(x-b)(x-c)(x-d)}. \] Hence or otherwise show that \begin{align*} &\frac{(a-\alpha)(a-\beta)(a-\gamma)(a-\delta)}{(a-b)(a-c)(a-d)} + \frac{(b-\alpha)(b-\beta)(b-\gamma)(b-\delta)}{(b-a)(b-c)(b-d)} + \text{two similar terms} \\ &= a+b+c+d-\alpha-\beta-\gamma-\delta. \end{align*}

Prove that if \[ y^2+z^2+yz=a^2, \quad z^2+zx+x^2=b^2, \quad x^2+xy+y^2=c^2, \quad yz+zx+xy=0, \] then \[ a\pm b\pm c=0. \]

Prove that if \(n\) is any integer, \[ \sin n\theta = 2^{n-1} \sin\theta \sin\left(\theta+\frac{\pi}{n}\right) \dots \sin\left(\theta+\frac{n-1}{n}\pi\right). \] Deduce that if \(n\) is any odd integer, the sum of the products taken two at a time of the \(n-1\) expressions \[ \tan^2\frac{\pi}{n}, \tan^2\frac{2\pi}{n}, \dots, \tan^2\frac{n-1}{2n}\pi, \] is \(\frac{1}{2}n(1-n)\).