Calendar: 1876-1877 Page 581

GENERAL LITERATURE AND SCIENCE 581 14 Find the polar equation to the parabola the vertex being the pole Two straight lines are drawn from the vertex of parabola at right angles to each other the points where these straight lines meet the curve are joined thus forming right-angled triangle find the least area of this triangle 15 Trace the curve χ Xs Find its equation when the origin is transferred to the vertex 16 Ρ is point on the parabola the focus find the equa- tion to the circle described on SP as diameter Shew that this circle touches the tangent at the vertex of the parabola 17 If BC be the semi-minor axis of the ellipse then BC2 CA2 CS and if SL be the semi-latus rectum SL AC Also if SBS' be right angle CA 2C& 18 If the tangent to the ellipse at any point Ρ intersect the directrix in the point and if be the focus corresponding to the directrix on which Ζ is situated then SZ will be at right angles to SP If NP produced meet the tangent at the extremity of the latus rectum in then QN PS 19 If the tangent at Ρ meet the axis major produced in and PN be the ordinate of the point then CF CN CA IfPThe tangent to the ellipse meeting the axis in and AP A'Ρ be produced to meet the perpendicular to the major axis through Τ in and then QT Q'T 20 If PN be the ordinate of any point Ρ on the ellipse then PA AN A'N BC1 AC To any point on the ellipse draw AQ A'Q meeting NP in and then NR NS PN 21 If from the foci Saud S' ST and ST are drawn at right angles to the tangent at then and Y' are on the circum- ference of the auxiliary circle and SY S'Y' BC2 If PN be the ordinate of then NY NY' PY PY' 22 The area of the ellipse is to the area of the auxiliary circle as BC to AC 23 Find the equation to the tangent at any point of an ellipse in terms of the tangent of its angle of inclination to the axis