\(A\) is a point on a fixed straight line \(l\) and \(B\) is a point on a second fixed straight line \(l'\) which intersects \(l\) in \(O\). \(C\) is the point between \(A\) and \(B\) dividing \(AB\) in the constant ratio \(\lambda:1\). If \(AB\) is of fixed length \(a\), prove that the locus of \(C\) is an ellipse whose area is independent of the mutual inclination of \(l\) and \(l'\), and show that this area cannot exceed \(\frac{\pi}{4} a^2\) whatever the value of \(\lambda\).
\(A\) and \(B\) are two fixed points which are conjugate with respect to a given conic \(S\). \(P\) is a variable point on \(S\), and \(PA\) and \(PB\) are drawn to cut \(S\) again in \(X\) and \(Y\) respectively. Prove that \(XY\) passes through a fixed point.
Prove that a sphere inverts into a sphere or a plane. Prove that if a sphere and two points \(P, P'\) inverse with respect to it, are inverted with respect to a point \(O\) not on the sphere, a sphere is obtained with the points \(Q, Q'\), into which \(P, P'\) will invert, as inverse points. What happens if \(O\) does lie on the sphere? Prove further that the inverse of a circle with respect to a point not in its plane is a circle.
The normals to a parabola at points \(A, B, C\) are concurrent in \(P\). If \(P\) lies on a fixed straight line, prove that the loci of the orthocentre and circumcentre of the triangle \(ABC\) are straight lines.
If \(A, B, C\) are points common to two rectangular hyperbolas which cut the circumcircle of \(ABC\) again in \(D_1\) and \(D_2\) respectively, prove that the distance between \(D_1\) and \(D_2\) is twice the distance between the centres of the two hyperbolas.
Prove that the midpoints of parallel chords of a conic are collinear. Find the equation of the pair of conjugate diameters common to the two conics given by \begin{align*} x^2+3xy+y^2+8x+7y &= 0, \\ 3x^2+4xy-2y^2+14x-4y+3 &= 0. \end{align*}
Prove Pascal's Theorem concerning the sides of a hexagon inscribed in a conic. Establish a ruler construction to obtain the asymptotes of a hyperbola, when three points of the curve are given and the directions of both asymptotes are known.
By projecting the theorem that the inscribed and escribed circles of a triangle touch the nine-point circle, deduce a theorem concerning all conics touching any three joins of four coplanar points \(A, B, C, D\), and passing through two particular points on a given straight line.
Prove that the roots of the equation \[ x^4 - x^3\left(4R+2\frac{\Delta}{s}\right) + x^2s^2 + x^2\frac{\Delta}{s}(4R+\frac{\Delta}{s}) - 2s\Delta x + \Delta^2 = 0 \] are the radii of the inscribed and three escribed circles of a triangle whose area is \(\Delta\), circumradius \(R\), and the sum of whose sides is \(2s\).
By considering the expression for \(\cos 7\theta\) in terms of \(\cos\theta\), find the roots expressed in trigonometric form of the equation \[ 64x^6 - 112x^4 + 56x^2 - 7 = 0. \]