The parameter of any diameter, is equal to four times the distance from its vertex to the focus. Lot BAD be a parabola, of which F is the focus, AC is any diameter, and BD its parameter; then is BD equal to four times AF. Draw the tangent AE; then, since AEFC is a parallelogram, AC is equal B to EF, which is equal to AF (Prop. IV.). Now, by Prop. IX., BC is equal to 4AF XAC; that is, to 4AF. Hence BC is equal to twice AF, and BD is equal to four times AF Therefore, the parameter of any diameter, &c. Cor. Hence the square of an ordinate to a diameter, is equal to the product of its parameter by the corresponding abscissa. 1 A If a cone be cut by a plane parallel to its side, the section is a parabola. Let ABGCD be a cone cut by a plane VDG parallel to the slant side AB, then will the section DVG be a parabola. Let ABC be a plane section through the axis of the cone, and perpendicular to the plane VDG; then VĖ, which is their common section, will be parallel to AB. Let bgcd be a plane parallel to the base of the cone; the intersection of this plane with the cone will be a circle. Since the B plane ABC divides the cone into two equal parts, BC is a diameter of the circle BGCD, and bc is a diameter of the circle bgcd. Let DEG, deg be the common sections of the plane VDG with the planes BGCD, bgcd respectively. Then DG is perpendicular to the plane ABC, and, consequently, to the lines VE, BC. For the same reason, dg is perpendicular to the two lines VE, bc. G Now, since be is parallel to BE, and A bB to eЕ, the figure bBEe is a parallelogram, and be is equal to BE.But because the triangles Vec, VEC are similar, we have ec : EC :: Ve : VE; and multiplying the first and second terms b of this proportion by the equals be and BE, we have be X ec : BE XEC :: Ve : VE. BA But since bc is a diameter of the circle bgcd, and de is perpendicular to bc (Prop. XXII., Cor., B. IV.), be Xec =de. For the same reason, BE X EC=DE. Substituting these values of be X ec and BE XEC in the preceding proportion, we have de : DES :: Ve: VE; that is, the squares of the ordinates are to each other as the corresponding abscissas ; and hence the curve is a parabola, whose axis is VE (Prop. VIII., Cor. 1.). Hence the parab. ola is called a conic section, as mentioned on page 177. G Every segment of a parabola is two thirds of its circum scribing rectangle. N E Let AVD be a segment of B K A a parabola cut off by the M straight line AD perpendicular to the axis; the area of AVD is two thirds of the circumscribing rectangle ABCD. G L Draw the line AE touching the parabola at A, and meeting the axis produced in E; and take a point H in the curve, so near to A that the tangent and curve may be regarded as coinciding. Through H draw KL perpendicular, and MN parallel to the axis. m'hen the rectangle AL : rectangle AM :: AGXGL: ABXAN :: AG XGE: ABX AG because GL or NH : AN::GE: AG. But GE is equal to twice GV or AB (Prop. V.); hence AL: AM :: 2:1; that is, AL is double of AM. Hence the portion of the parabola included between two ordinates indefinitely near, is double the corresponding portion of the external space ABV. Therefore, since the same is true for every point of the curve, the whole space AVG is double the space ABV. Whence AVG is two thirds of ABVG; and the segment AVD is two thirds of the rectan- gle ABCD. Therefore, every segment, &c. B 1. An ellipse is a plane curve, in which the sum of the distances of each point from two fixed points, is equal to a given line. 2. The two fixed points are called the foci. Thus, if F, F are two fixed points, and if the point D moves about F in such a manner that the sum of its distances from F and F is always the same, the point D will describe an F F ellipse, of which F and F are the foci. 3. The center is the middle point of the straight line joining the foci. 4. The eccentricity is the distance from the center to either focus. Thus, let ABA'B' be an ellipse, F and Fl the foci. Draw the line FFi and bisect it in C. The point C is the center of the ellipse; and A A CF or CF' is the eccentricity. 7 5. A diameter is a straight line drawn through the center, and D' terminated both ways by the B curve. 6. The extremities of a diameter are called its vertices. Thus, through C draw any straight line DD' terminated by the curve; DD' is a diameter of the ellipse ; D and D' are its vertices. 7. The major aris is the diameter which passes through the foci ; and its extremities are called the principal vertices. 8. The minor axis is the diameter which is perpendicular to the major axis. Thus, produce the line FF to meet the curve in A and Al; and through C draw BB' perpendicular to AA'; then is AA' the major axis, and BB! the minor axis. 9. A tangent is a straight line which meets the curve, but, being produced, does not cut it. C K G 10. An ordinate to a diameter, is a straight line drawn from any point of the curve to the diameter, parallel to the tangent at one of its vertices. Thus, let DD be any diameter, T and TT' a tangent to the ellipse at D. From any point G of the EG curve draw GKG parallel to TT T and cutting DD in K; then is R It is proved in Prop. XIX., Cor. D 1, that GK is equal to G'K; hence the entire line GG' is called a double ordinate. 11. The parts into which a diameter is divided by an ordinate, are called abscissas. Thus, DK and D'K are the abscissas of the diameter DD corresponding to the ordinate GK. 12. Two diameters are conjugate to one another, when each is parallel to the ordinates of the other. Thus, draw the diameter EE' parallel to GK, an ordinate to the diameter DD', in which case it will, of course, be parallel to the tangent TT'; then is the diameter EE' conjugate to DD. 13. The latus rectum is the double ordinate to the major axis which passes through one of the foci. Thus, through the focus F T'. draw LL a double ordinate to L D the major axis, it will be the latus rectum of the ellipse. 14. A subtangent is that part F G of the axis produced which is included between a tangent and the / ordinate drawn from the point of contact. Thus, if TT be a tangent to tre curve at D, and DG an ordinate to the major axis, then GT is the corresponding subtangent. |