Calendar: 1870-1871 Page 509

GENERAL LITERATURE AND SCIENCE 509 12 Find the equation to the tangent to the parahola at the point and prove that the coordinates of the points of inter- section of tangents at h' k' are h' and 2a 13 Find the equation to the normal to parabola in terms of the angles which it makes with the axis Two equal parabolas have the same focus and their axes at tight angles to each other and normal to one of them is per- pendicular to normal to the other prove that the locus of the intersection of such normals is parabola Ρ is normal chord at an point Ρ of parabola If θ and φ be the angles the tangents at Ρ and make with the axis tan θ φ tun Θ 14 Find the equation to the pair of lines drawn from the centre of an ellipse to the points where it is intersected by the itraight line whose equation is Ix -f- my Hence deduce the condition that the given line may touch the ellipse If two equal and similar ellipses have the same centre show that their points of intersection are at the extremities of diameters at right angles to one another 15 The chord of contact of two tangents to parahola sub- tends an angle β at the vertex show that the locus of their point of intersection is an hyperbola whose asymptotes are in- clined to the axis of the parabola at an angle φ such that tan φ tan β 16 Show that the equation to parabola referred to pair of tangents as axes mnv be put into the form Find also the equation to the circlo in an analogous shape How many varieties of sign may be taken and what is repre- senled by each 17 The coordinates of the foci of the conic ax1 42 bxy cy- are given by the equations י