\(ABCD\) is a square of side \(a\). A point \(P\) moves so that the sum of the squares of its distances from \(A, B, C, D\) is equal to \(3a^2\). Prove that its locus is a circle and find the radius.
Given one vertex \(A\), the circumcentre \(O\), and the orthocentre \(H\) of a triangle, show how to construct the triangle. Two triangles \(ABC\) and \(AB'C'\) with common vertex \(A\) are such that the orthocentre of each is the circumcentre of the other. Prove that the point of intersection of \(BC\) and \(B'C'\) is equidistant from the circumcentre and orthocentre.
If a circle \(S\) when inverted with respect to a circle \(\Sigma\) becomes a circle \(S'\), show that \(S, \Sigma,\) and \(S'\) are members of a coaxial system. Prove that for any triangle the circumcircle, the nine-points circle, and the polar circle (that is, the circle with respect to which the triangle is self-polar) are coaxial.
Establish Pascal's theorem that if a hexagon is inscribed in a conic then the meets of the three pairs of opposite sides are collinear. State the dual theorem. Prove that if a tangent to a hyperbola at a point \(P\) meets the asymptotes in \(A\) and \(B\), then \(AP=PB\).
\(P\) and \(Q\) are two points on an ellipse of which \(S\) is a focus and are such that \(\angle PSQ\) is always a constant angle. Show that the tangents at \(P\) and \(Q\) meet on a fixed conic, and that the chord \(PQ\) envelops a second fixed ellipse. In what circumstances can the former conic be a hyperbola?
Prove that the two pairs of lines \(ax^2+2hxy+by^2=0\) and \(a'x^2+2h'xy+b'y^2=0\) are harmonically conjugate if \(ab'+a'b=2hh'\). In a given plane \(A, B\) are two fixed points and \(l, m\) two fixed lines. A point \(P\) is such that \(PA, PB\) harmonically separate the lines through \(P\) parallel to \(l, m\). Show that the locus of \(P\) is a hyperbola whose asymptotes are parallel to \(l, m\), and that its centre lies on \(AB\).
\(P\) is a point on a conic \(S=0\) and the tangent at \(P\) has equation \(T=0\) while \(L=0\) represents any line through \(P\). Interpret the equations:
Obtain the conditions for the general equation of the second degree \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] to be an ellipse, a parabola, or a hyperbola. Show that if the equation represents a rectangular hyperbola then referred to the asymptotes as co-ordinate axes \(O'\xi, O'\eta\), its equation will be \[ 2(h^2-ab)\xi\eta = \Delta, \] where \(\Delta = \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix}\).
(i) Eliminate \(\theta\) from the equations \[ a\tan\theta+\sec\theta=h, \quad a\cot\theta+\csc\theta=k. \] (ii) Sum the infinite series \[ \frac{1}{2!} + \frac{4}{3!} + \frac{9}{4!} + \dots + \frac{n^2}{(n+1)!} + \dots. \]
If \(\theta=2\pi/7\), prove that \begin{align*} \sin\theta+\sin2\theta+\sin4\theta &= \sqrt{7}/2, \\ \tan^2\theta+\tan^2 2\theta+\tan^2 4\theta &= 21. \end{align*}