55. If from any point T on a fixed tangent a second tangent TP be drawn, the angle STP will be constant.

56.

The vertices of any circumscribed triangle are concyclic with the focus.

57. PQR being a circumscribed triangle, the perpendiculars from P, Q, R to SP, SQ, SR cointersect.

58. If the tangents at P, Q meet in T, and if C be the centre of the circle TPQ, then CST is a right angle.

59. The normal at P bisects the angle between SP and the diameter through P.

60. Two parabolas which have a common focus and their axes in opposite directions intersect at right angles.

61. The perpendicular drawn to a normal from the point in which it meets the axis envelopes an equal parabola.

62. Normals at the extremities of a focal chord intersect on the diameter which bisects the chord.

63. If PQ be a focal chord, and R the foot of the perpendicular upon it from the intersection of the normals at P, Q, then SP = QR.

64. The portion of any tangent intercepted by the tangents at fixed points P, Q subtends a constant angle at S. The angle subtended is a right angle when PQ passes through S.

65. If a circle through S touches the parabola in P, Q, then SP is equal to the latus rectum.

66. The normal at any point is equal to twice the focal perpendicular upon the tangent, and is also a mean proportional between the focal distance of that point and the latus rectum.

67. The squares of the normals at the ends of a focal chord are together equal to the square of twice the normal perpendicular to the chord.

68. The locus of the vertex of a parabola which has a given focus and touches a given straight line is a circle.

69. The circle on a focal radius touches the tangent at A.

70. The tangent and normal at any point are bisected by the focal perpendiculars upon them; and the straight line joining the feet of the perpendiculars is parallel to the axis.

71. The diameter through one end of a focal chord bisects the chord normal at the other.

72. The locus of the foot of the focal perpendicular on the normal is a parabola.

73. If the tangents at P, Q intersect in R, the circle through P touching QR in R passes through S.

74. If from the foot of the normal at P a perpendicular PK be drawn to SP, then PK = 2AS.

75. Determine the position of P so that the triangle SPG may be equilateral.

76. The tangent at any point meets the directrix and the latus rectum in points equidistant from the focus.

77. If QQ' be the focal chord perpendicular to the normal at P, then PG=SQ. SQ.

78. The triangle bounded by three tangents to a parabola is equal to half the triangle whose vertices are at the points of contact.

79. If two parabolas be described each touching two sides of a given equilateral triangle at the points in which it meets the third side, prove that they have a common focus and that the tangent to either of them at their point of intersection is parallel to the axis of the other.

80. Two equal parabolas have the same axis and directrix. From a point on one of them tangents are drawn to the other. Shew that the perpendicular from that point to the chord of contact is bisected by the axis.

81. Supposing the triangle 123 in Ex. 42 to become. evanescent, shew that in the limit the common chord of the circle and the parabola is equal to four times their common tangent measured from the curve to the axis.

82. A diameter meeting a chord and the tangent at an end of it is cut by the curve in the ratio in which it cuts the chord. 83. Draw a chord which shall be cut in a given ratio by a given diameter.

84. A parabola being inscribed in a triangle its directrix passes through the orthocentre.

CHAPTER III.

CHORD-PROPERTIES OF THE ELLIPSE.

40. DEF. A Conic is the curve described in a plane by a point which moves in such a way that its distance from a certain fixed point, called the Focus, is in a constant ratio to its perpendicular distance from a certain fixed straight line, called the Directrix.

This constant ratio is called the Eccentricity.

A Conic is called an Ellipse, a Parabola, or a Hyperbola, according as its eccentricity is less than, equal to, or greater than unity.

41. Some properties can be proved as simply for all conics at once as for the ellipse separately. We shall accordingly, in enunciating some of the propositions in Chapters III. and IV., use or imply the general term Conic instead of Ellipse. Articles which apply to all conics will be distinguished by

the mark T.

42. Let S be the focus of an ellipse, and X the point in which the axis meets the directrix. Divide SX in A so that SA may be to AX as the eccentricity. Then A is a vertex. Since the eccentricity is less than unity it is evident that there is a second vertex A' in XS produced, such that SA' is to A'X as the eccentricity.

43. The ellipse lies wholly on the same side of the directrix with the focus. For imagine two of its points O, P to lie on opposite sides. Let OP cut the directrix in R. Then,

by the definition if OP be at right angles to the directrix, and otherwise a fortiori,

By addition, 80+ SP OP, which is impossible. Hence the ellipse lies wholly on one side of the directrix, viz., that on which are the points A, A' already determined.

¶44. An extension of the definition of a conic.

From any point P on a conic draw PM perpendicular to the directrix, and PR meeting the directrix at any constant angle. Then PM: PR is a constant ratio. But by definition SP: PM is constant. Therefore SP: PR is constant. Hence a conic might have been defined as the locus of a point P whose distance SP from the focus is in a constant ratio to its distance PR from the directrix measured parallel to any fixed straight line which meets the directrix. When this fixed straight line meets the directrix perpendicularly we come back to the original definition.

T45. As a particular case of the above let P, Q be two points on a conic*, and let the straight line joining them meet the directrix in R. Let fall perpendiculars PM, QN on the directrix.

Then

SP: SQ=PM: QN
= PR: QR,

[Def.

by parallels, and conversely if this relation holds and P be on the curve, then Q will be on the curve.

*This includes the case of a focal chord.

46. For two chords of an ellipse through P and A, A' respectively which meet the directrix in Z, Z',

147. From Art. 45, when one point P on the curve is given a second Q may be determined by drawing PR to any point R on the directrix, and then drawing SQ making the same angle as PS with SR. For SQ will meet RP in a point Q such that

SP: SQ = PR : QR.

[Euc. VI. A.

From this construction it appears that a straight line which meets a conic in a point P will in general meet it in one other point Q, and that no straight line can meet a conic in more points than two. Hence Conics are called curves of the second degree.

48. Starting from the vertex A we may determine any number of points as P on the ellipse, by drawing AZ to any point Z on the directrix, and then making the angle ZSP equal to the known angle XSZ. The line pS thus drawn meets ZA on the ellipse.

[47.

Or we might have determined points as P by starting from the vertex A' and taking the angle Z'SP equal to XSZ', where Z' is any assumed point on the directrix.