. Plane and solid analytic geometry; an elementary textbook. en that FD = DF = m, OD = = BC = m 1 -e2 CA-- = A0 = a em 1-e2 nn - - wr? l em It follows that CD = -, and that the equations of thedirectrices are x = % and sc = -^Le e [38] Ch. IX, § 65) CONIC SECTIONS 109 Also that OF=CD-FD = -^— - m = -tS!L. = ae. 1 — e1 1 — e2 It is convenient to let OF be represented by a singleletter c. Then c = ae, or e = - In obtaining equation [36], we let b2 = a2(l — e2).Solving for e2, we have e,=q^J>*t |-39-j Comparing these two values of e, we havea2 — 62 = c2. [40] From this we see that BF, being th
. Plane and solid analytic geometry; an elementary textbook. en that FD = DF = m, OD = = BC = m 1 -e2 CA-- = A0 = a em 1-e2 nn - - wr? l em It follows that CD = -, and that the equations of thedirectrices are x = % and sc = -^Le e [38] Ch. IX, § 65) CONIC SECTIONS 109 Also that OF=CD-FD = -^— - m = -tS!L. = ae. 1 — e1 1 — e2 It is convenient to let OF be represented by a singleletter c. Then c = ae, or e = - In obtaining equation [36], we let b2 = a2(l — e2).Solving for e2, we have e,=q^J>*t |-39-j Comparing these two values of e, we havea2 — 62 = c2. [40] From this we see that BF, being the hypotenuse of aright triangle whose legs are c and 6, is equal to a. Italso shows that a isalways larger than 5, orthat AA(=2a), theaxis perpendicular to thedirectrices, is larger thanBB(=2b). AA hasbeen called the trans-verse or principal axis;BB is called the conju-gate or minor axis of thecurve. If the foci of the ellipseare on the F-axis, thevertex A also lies on that axis, and B on the X-axis (Fig. 60). Its equation is (seeend of Art. 64). V + ^=1. [41] 110 ANALYTIC GEOMETRY [Ch. IX, § 66 All the formulas found above hold for [41], except theequations of the directrices, which are »-*r PROBLEMS 1. Find a, b, c, e, and the equations of the directrices in theellipse, (a) 4a2 + 9/ = 36, , (b) 9a2 + 4/ = 36, (c) 3 ar + $ if = 10. 2. Find the equation of the ellipse having its centre at theorigin and its foci on the X-axis, if (a) a? = 3 and b = 2, (d) b = 4 and c = 3, (6) 6 = 3 and e = i (e) a = 5 and c = 3, (c) a = 6 and e = f, (/) c = 4 and e = -|. 3. Show that the length of the latus rectum (line through 2 b2the focus perpendicular to the axis) of the ellipse is 4. Show that the circle is the limiting form of the ellipseas a and b approach equality. What is the eccentricity of thecircle, and where are its foci and directrices ? 5. What is the equation of the ellipse which has its centreat the origin and its axes coincident with the coordinate axes,and which p
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