Two points \(X, Y\) of general position are taken in the plane of a fixed circle \(C\). Obtain a construction for another circle passing through \(X\) and \(Y\) and intersecting \(C\) at the ends of a diameter. Justify the construction.
The sides \(AB, BC, CD,\) and \(DA\) of a skew quadrilateral are cut by a plane in the four points \(P, Q, R,\) and \(S\) respectively. Show that \[ AP \cdot BQ \cdot CR \cdot DS = AS \cdot BP \cdot CQ \cdot DR, \] and establish the converse theorem. If a sphere touches internally the sides of a skew quadrilateral, show that the four points of contact lie on a circle.
Prove that the fixed line \(lx+my+1=0\) bisects the chord of the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] that lies along the line \(px+qy+1=0\) if \[ a^2p(p-l) + b^2q(q-m)=0. \] Hence show that the lines containing such chords envelop a parabola with directrix \[ a^2lx+b^2my+a^2+b^2=0. \]
Show that with a suitable choice of pole and initial line the equation of a conic in polar coordinates is of the form \[ l=r(1+e\cos\theta). \] Show that the locus of the point from which the tangents to this conic are perpendicular has equation \[ r^2(e^2-1) - 2elr\cos\theta + 2l^2 = 0, \] and state the nature of this locus.
Find the tangential equation of the general conic \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] referred to rectangular axes. Show that the foci of the conic are at the intersections of the two rectangular hyperbolas \begin{align*} C(x^2-y^2)-2Gx+2Fy+A-B &= 0, \\ Cxy-Fx-Gy+H &= 0, \end{align*} where \(A, B, C, F, G, H\) are the coefficients in the tangential equation of the conic. Find the focus and directrix of the parabola \[ 4x^2-4xy+y^2+22x-6y+24=0. \]
The pair of tangents from the point \((t^2,t,1)\) of the conic \(y^2=zx\) to the conic \[ ax^2+by^2+cz^2=0 \] meet the first conic again in \(P\) and \(Q\). Find the equation of \(PQ\). Hence show that if \(b^2+ca=0\) there are an infinite number of triangles inscribed in the first conic and circumscribing the second.
Three points and an asymptote of a hyperbola (but not the curve itself) are given. Obtain and justify a construction for the other asymptote.
Four conics pass through four common points \(A,B,C,D\). Prove that if the four tangents to them at \(A\) form a harmonic pencil, then so does the pencil of tangents at any other of the points \(B,C,\) and \(D\).
Determine numbers \(A,B,\) and \(C\) such that for all \(\theta\) \[ A\sin^5\theta + B\sin^3\theta + C\sin\theta = \sin 5\theta. \] Hence show that \(\sin \pi/30\) is a root of the equation \[ 16x^4+8x^3-16x^2-8x+1=0, \] and give the remaining roots as sines of angles.
The circumcircle of an obtuse angled triangle \(ABC\) subtends an angle \(2\theta\) at the orthocentre. Show that \[ 8\cos A \cos B \cos C + \cot^2\theta = 0. \] Show also that \(\theta\) is never less than \(\sin^{-1}\frac{1}{3}\).