\(P\) is any point on the circumcircle of a triangle \(ABC\) and \(A', B', C'\) are the other ends of the diameters through \(A, B, C\) respectively; if \(PA', PB', PC'\) meet \(BC, CA, AB\) respectively in \(X, Y, Z\), prove, by using Pascal's Theorem or otherwise, that \(X, Y, Z\) lie on the same diameter of the circumcircle.
The lines joining the vertices \(A, B, C\) of a triangle to a point \(P\) cut the opposite sides in \(L, M, N\) respectively; if \(X, Y, Z\) are the middle points of \(MN, NL, LM\) respectively, prove that the lines \(AX, BY, CZ\) are concurrent.
Prove that the foot of the perpendicular from the focus of a parabola to a variable tangent lies on a fixed line. \par If \(S\) is a focus of an ellipse, prove that the parabola with its focus at \(S\) touching the tangent and normal at a point of the ellipse also touches the minor axis of the ellipse.
\(BC, AD\) are two chords of a conic through a focus \(P\) of the conic; if \(CA, BD\) meet at \(Q\) and \(AB, CD\) meet at \(R\), prove that \(QR\) subtends a right angle at the focus \(P\).
Prove that the inverse of a circle with respect to a sphere is in general another circle, but may specially be a straight line. \par The lines through a fixed point \(O\) and the points of a fixed circle \(\Gamma\) generate a cone; prove that any plane parallel to the plane of \(\Gamma\) cuts the cone in a circle and that there is another set of parallel planes cutting the cone in circles. \par Find the relation between \(O\) and \(\Gamma\), if the two sets of parallel planes are the same.
Prove that each of the pairs of lines \(ax^2 + 2hxy + by^2 = 0\), \(px^2 + 2qxy + ry^2 = 0\) is harmonically separated by the pair of lines \[ \begin{vmatrix} ax+hy & hx+by \\ px+qy & qx+ry \end{vmatrix} = 0. \] Shew that \((a-b)(p-r)+4hq=0\) is the necessary and sufficient condition that one of the angle bisectors of \(ax^2+2hxy+by^2=0\) should make an angle of \(\pi/4\) with one of the angle bisectors of \(px^2+2qxy+ry^2=0\), when the coordinate axes are perpendicular.
The coordinates of the vertices of a triangle referred to rectangular axes are \((R \cos\alpha, R\sin\alpha)\), \((R\cos\beta, R\sin\beta)\), \((R\cos\gamma, R\sin\gamma)\); find the coordinates of (i) the circumcentre, (ii) the centroid, (iii) the orthocentre.
The common points of the two rectangular hyperbolas \begin{align*} ax^2 + 2hxy - ay^2 + px + qy &= 0, \\ 2h'xy + p'x + q'y &= 0 \end{align*} are at the origin of coordinates and at the points \(L, M, N\); prove that the equation of the circle through \(L, M, N\) is \[ (ax+hy+p)(h'x+q') - (hx-ay+q)(h'y+p') = 0. \]
Shew that the foci of the central conic \(ax^2 + 2hxy + by^2 + c = 0\) are given by \[ \frac{x^2-y^2}{a-b} = \frac{xy}{h} = \frac{c}{ab-h^2}, \] the coordinate axes being rectangular. \par Find the coordinates of the real foci of the conic \[ 5x^2+4xy+2y^2-42x-24y+105=0. \]
The lines joining the vertices of a triangle \(XYZ\) to a point \(P\) cut the opposite sides in \(L, M, N\) and \(s\) is any conic through the four points \(P, L, M, N\). If this conic \(s\) cuts \(YZ\) again at \(A\) and if \(A'\) is the harmonic conjugate of \(A\) with respect to \(Y, Z\), prove that \(A'P\) is the tangent to \(s\) at \(P\).