A B meters of the diameters, to which they are ordinates, by P and P'; and L a line drawn from their point of intersection to the curve parallel to the axis ; then (I. 10) R:R' =P:L:P.L=P:P (Pl. Ge. VI. 1). PROPOSITION XIII. If from any point in a parabola, an ordinate be drawn to a diameter, and a tangent to the curve from the same point ; and if a circle be described, having the ordinate for a tangent, and the fourth part of the parameter of the diameter through the given point as a cord; a diameter to the parabola passing through the point of intersection of the tangent and the circle, will cut off a fourth part of the ordinate Let EH be a tangent at the point E, and EG an ordinate to the diameter HM of a parabola ; EIK a circle, having EK for a cord, and EG for a tangent; and IL a diameter through I; then EG = 4EL. The two triangles EKI, EHM, are similar, because the alternate angles KEI, EHM, are equal, and angles EKI, HEM, are equal (Pl. Ge. III. 32); therefore KE: IE = EH : MH, or KE: IE=EH: 2VH (I. 8). Therefore 2KE: VH=IE-EH, but 4KEEN = NV2 (I. 7) = EH?; and hence 21È · EH = EH? = EH · EH; and therefore 21E = EH. Since EH is bisected in I, therefore EL = LM, or 4EL = EG. EXERCISES. 1. Any straight line, drawn from the focus of a parabola to a point in the directrix, is a mean proportional between half the parameters of the diameters which pass through its extremities. 2. If from any point in the parabola, a tangent, semiordinate, and perpendicular, be drawn to meet the same diameter, their squares will be to one another as the parameters of three diameters, that which passes through the point of contact, that which they meet, and the axis. 3. If through the focus of a parabola a semi-ordinate be applied to the axis, and from its extremity a tangent be drawn to meet another semi-ordinate produced ; then shall the produced semi-ordinate be equal to the line joining its extremity in the curve and the focus. 4. If a tangent be drawn from any point in the curve, to meet the axis produced, and from the point of contact a perpendicular to the tangent, and a semi-ordinate to the axis, be drawn; then the segment of the axis between the perpendicular and semi-ordinate will be equal to half the parameter of the axis, and the segment between the perpendicular and tangent equal to half the parameter of the diameter which passes through the point of contact. 5. To find the directrix and focus of a parabola given in position. 6. If from the vertex of any diameter, a straight line be drawn to the extremity of a semi-ordinate so as to meet another semi-ordinate, the latter will be a mean proportional between its segments next the diameter and the former. 7. A diameter of a parabola, the tangent at the vertex of that diameter, and a point in the curve, being given, to find the directrix and focus. 8. If from any point in a tangent, a parallel to the diameter passing through the point of contact be drawn to meet an ordinate of the same, the rectangle under the parallel, and its external segment, will be equal to the square of that part of it that is between the curve, and the line joining the vertex of the diameter, and either extremity of its ordinate. 9. The focus and directrix of a parabola being given, to draw a tangent to the curve parallel to a line given in position that is not perpendicular to the directrix. 10. If there be two tangents to a parabola, such, that the straight line joining their points of contact pass through the focus, they will cut each other at right angles, their intersection will be in the directrix, and the line joining it with the focus will be perpendicular to the line joining the points of contact. 11. From a given point, in a given parabola, to draw a tangent without finding the focus. 12. Two parabolas of equal parameters, having their axes in the same line, and in the same direction, but their vertices at different points, being produced indefinitely, continually approach, but never meet. 13. In a given parabola to find a diameter that makes a given angle with its ordinates. 14. If from any point in a tangent, a straight line be drawn, to meet the curve and the diameter passing through the point of contact, the square of its segment, between the tangent and the diameter, will be equal to the rectangle under its segments between the tangent and the points in the curve. 15. The focus and directrix of a parabola being given, to draw a tangent to the curve from a given point without it.' 16. If from a point in a parabola, a semi-ordinate be applied to a diameter, and from the same point any other straight line be drawn to meet it, the square of this line will be equal to the rectangle under the absciss of that diameter, and the parameter of the diameter to which the straight line, when produced, is an ordinate. 17. If from points in the curve, two tangents be drawn to meet, their squares will be to each other as the parameters of the diameters passing through the points of contact. 18. If two tangents, and the line joining their points of contact, meet the same diameter, the segment of the diameter, intercepted by this line from the vertex, will be a mean proportional between those intercepted by the tangents. SECOND BOOK. ELLIPSE. DEFINITIONS. 1. If two points be given in position, the locus of a point, the sum of whose distances from them is always the same, is a curve called an ellipse. 2. The given points are named the foci, and the middle of the line that joins them, the centre of the ellipse. . 3. The distance of the centre from one of the foci is called the eccentricity. "J' V 'Ini! Is i peli? . .' . 4. A diameter is a straight line drawn through the centre, and terminated on both sides by the curve. 5. The diameter which passes through the foci is named the transverse or major axis, and that which is perpendicular to it, the conjugate or minor axis. 6. An ordinate to a diameter is a straight line not passing through the centre, but terminated by the curve, and bisected by the diameter. 7. Two diameters are said to be conjugate to one another when each is parallel to the ordinates of the other. 8. The parameter of a diameter is a third proportional to that diameter and its conjugate. PROPOSITION I. If from any point in an ellipse, straight lines be drawn to the foci, their sum is equal to the transverse axis. Let EFGH be an ellipse, of which EF is the transverse, and GH the conjugate axis, GDP For EA+EB=FA +FB; EFA lines be drawn to the foci, from a point without the ellipse, their sum is greater than the transverse ; but, if from a point within it, less. For AP + PB = AD + DP + PB. But DP + PB > DB; therefore AP + PB 7 AD + DB, or AP + PB 7 EF. And it is similarly shown that AQ +QB < EF. Cor. 2.-A point is without, in, or within the curve, ac cording as the sum of the lines drawn to the foci is greater, equal, or less, than the transverse axis. COR. 3.-The distance of either extremity of the conju gate axis, from one of the foci, is equal to half the transverse. For if GB be joined, the triangles ACG, BCG, are equal (II. Def. 2 and 5, and Pl. Ge. I. 4); therefore AG = GB. |