Plane and solid analytic geometry; an elementary textbook . tan1! and there will be a real value of p, these val-ues growing larger as 6 approaches tan_1(H—j 112 ANALYTIC GEOMETRY [Ch. IX, § 66 tan-1( ). The lines then which pass through the V aJ f V\ f V\origin, making the angles tan M H J and tan_1( J with the JT-axis, do not cut the curve, while every linelying between these lines cuts the curve in two real curve must therefore approach parallelism with theselines as the point recedes from the origin, and it will beshown in the next article that the curve approaches coinci-dence
Plane and solid analytic geometry; an elementary textbook . tan1! and there will be a real value of p, these val-ues growing larger as 6 approaches tan_1(H—j 112 ANALYTIC GEOMETRY [Ch. IX, § 66 tan-1( ). The lines then which pass through the V aJ f V\ f V\origin, making the angles tan M H J and tan_1( J with the JT-axis, do not cut the curve, while every linelying between these lines cuts the curve in two real curve must therefore approach parallelism with theselines as the point recedes from the origin, and it will beshown in the next article that the curve approaches coinci-dence with these lines. Such a line is called an we continue to increase 0, we see that there will be no real value until tan 6 again becomes numerically b>less than — Then p goes through the same changesa in value, decreasing until it equals ± a. But we haveshown that the curve is symmetrical with respect to bothaxes, and there is therefore no need of discussion beyond thefirst quadrant. The following is the form of the hyperbola: 7. Place the points F and D on the X-axis so thatFC=OF md DC=CD, and draw DH perpendicular Ch. IX, § GG] CONIC SECTIONS 113 to the X-axis. The symmetry of the curve again shows,as in the ellipse, that the hyperbola may be said to havetwo foci, J7 and F\ and two directrices, DR and DW. We can now l>!lfc^tln relations between the variouslines in the figure. ^Bliavc seen that DF=FD = m, e*-r CA = AC =a = em e2-l CB = B0 =b =—— VeP-1 It follows that CD = -, and that the equations of thedirectrices are a t a ac = Also that * = 2, and * = -?. , [42] 0F= CD + DF=^-+m = -pL = — 1 e& — 1 It is convenient to let OF be represented by a singleletter c. Then c = ae, e or e = . a In obtaining equation [37], we let V2 = a2(e2 — 1). Solving for e2, we have Comparing the two values of e, we have a? + b2 = c2. [44] 114 ANALYTIC GEOMETRY [Ch. IX, § G7 There is, in the hyperbola, no restriction on the relativevalues of a and b.
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