Show that there are three normals from a given point to a parabola. If \(P\) is a point on a parabola whose vertex is \(A\), and \(Q, R\) are the feet of normals from \(P\) to the curve, show that as \(P\) moves along the curve, \(QR\) passes through a fixed point, and that \(AP\) and \(QR\) meet on a fixed line.
Ellipses are drawn through the middle points of the sides of the rectangle \[ (x^2-a^2)(y^2-b^2)=0. \] Find the general equation of the family, and show that there is a hyperbola having the diagonals as asymptotes which cuts all the ellipses orthogonally.
Find the magnitudes and directions of the axes of the conic \[ x^2+xy+y^2-2x+2y-6=0. \]
Prove that the coordinates of any point on the general conic can be expressed in terms of a parameter \(t\) in the form \[ x:y:1 :: a_1t^2+2b_1t+c_1 : a_2t^2+2b_2t+c_2 : a_3t^2+2b_3t+c_3 \] and find the condition that \(lx+my+n=0\) should touch the conic. Show that the equation of the director circle is \[ (a_1-a_3x)(c_1-c_3x) + (a_2-a_3y)(c_2-c_3y) = (b_1-b_3x)^2+(b_2-b_3y)^2. \]
Using areal or trilinear coordinates, find the coordinates of the centre of a conic circumscribing the triangle of reference. Two such conics touch at one of the angular points. Show that their centres, the point of contact and the middle points of the sides of the triangle lie upon a conic.
A rectangle is formed by drawing a pair of parallel lines through two given points A, B and a pair of parallel lines perpendicular to the former pair through two given points C, D. Prove that the locus of the centre of the rectangle is a circle which bisects AB and CD.
A variable point X is taken on the side BC of a triangle ABC in which AB=AC. Points Y, Z are taken in CA, AB such that CY=BX and BZ=CX. Prove that the triangle XYZ is of constant shape and that its circumcentre coincides with the in-centre of the triangle ABC.
Prove that the inverse with respect to the circumcircle of a triangle ABC of its nine-point circle is the circumcircle of the triangle formed by the tangents at A, B, C.
Prove that the mid-points of the six edges of a parallelopiped which do not pass through either of two given opposite corners of the parallelopiped are the vertices of a plane hexagon such that the lines joining opposite vertices meet in a point.
Prove that the circumcircle of the triangle formed by three tangents to a parabola passes through the focus. The tangents to a parabola at P and Q meet in T, and the tangent that is parallel to PQ cuts TP, TQ in X, Y. Prove that, if the circle TXY cuts the diameter through T in Z, then SZ is parallel to PQ.