Define harmonic conjugates. \(X, Y, Z\) are collinear points on the sides \(BC, CA, AB\) of a triangle and \(X', Y', Z'\) are their harmonic conjugates with respect to \(B, C\); \(C, A\); and \(A, B\) respectively. Prove that \(AX', BY', CZ'\) are concurrent.
Prove that, if through any point \(O\) chords \(OPP', OQQ'\) of a conic are drawn in fixed directions, the ratio \(OP.OP':OQ.OQ'\) is constant. Prove that two tangents from a point to a central conic are proportional to the parallel chords through the points of contact.
Prove that the reciprocal of a conic with respect to any circle, having its centre at a focus of the conic, is a circle. Shew that, if \(A, B, C, D\) be four given points and four conics are drawn with the same focus \(S\), each touching the sides of the triangle formed by three of the four points, then these four conics have a common tangent.
The vertices of a triangle lie on the lines \[ y=m_1x, \quad y=m_2x, \quad y=m_3x, \] and the orthocentre is at the origin. Prove that the locus of the centroid is the line \[ x(m_1+m_2+m_3+3m_1m_2m_3) = y(3+m_2m_3+m_3m_1+m_1m_2). \]
Find the equation to the normal at any point on the parabola \(x=am^2, y=2am\). Prove that perpendicular normals to this parabola intersect on the parabola \[ y^2 = a(x-3a). \]
Find the point of intersection of the normals to the ellipse \(b^2x^2+a^2y^2=a^2b^2\) at the ends of the chord \(lx+my=1\). If this point lie on the diameter which is conjugate to the chord, shew that the chord touches the ellipse \[ \frac{x^2}{a^4} + \frac{y^2}{b^4} = \frac{1}{a^2+b^2} \] or else that it is parallel to one of the axes.
Prove that two real conics of a confocal system pass through any point in their plane and that they cut at right angles. Prove that the centre of curvature of one of the conics at one of their common points is the pole with regard to the second conic of the tangent at that point to the first conic.
Find the centre and the lengths and directions of the axes of the curve \[ 17x^2+12xy+8y^2-32x+24y+64=0, \] and draw the curve.
Prove Pascal's Theorem that the three points of intersection of the opposite sides of a hexagon inscribed in a conic are collinear. \(AA', BB'\) are fixed straight lines through two fixed points \(A, B\); \(C, D\) are two other fixed points. A variable conic through \(A, B, C, D\) meets \(AA'\) in \(P\) and \(BB'\) in \(Q\); shew that \(PQ\) passes through a fixed point on \(CD\).
In the interior of the parallelogram \(ABCD\) a point \(P\) is taken such that the sum of the angles \(APB, CPD\) is two right angles; shew that \[ AP.CP + BP.DP = AB.BC. \]