. Plane and solid analytic geometry . 0); V2.(e) 50.^-62/-= 8.(d) 2jc^ - f, = 4. (/) 6 a;2 -9^/^ = 4. 2. If the eccentricity of a hyperbola is 2 and its major axisis 3, what is the length of its minor axis ? Ans. 3 V3. 3. How far apart are the foci of the hyperbola in Ex. 2 ? Ans. 6. 4. What is the equation of the hyperbola whose eccentricityi^ V2and whose foci are distant 4 from each other ? 4! • 1 5. The extremities of the minoraxis of a hypferbola are inthe points (0, ± 3) and the eccentricity is 2. Find the equa-tion of the hyperbola. w,/ - 7^. Jb s r i 6. Show that, in terms of a and h, e
. Plane and solid analytic geometry . 0); V2.(e) 50.^-62/-= 8.(d) 2jc^ - f, = 4. (/) 6 a;2 -9^/^ = 4. 2. If the eccentricity of a hyperbola is 2 and its major axisis 3, what is the length of its minor axis ? Ans. 3 V3. 3. How far apart are the foci of the hyperbola in Ex. 2 ? Ans. 6. 4. What is the equation of the hyperbola whose eccentricityi^ V2and whose foci are distant 4 from each other ? 4! • 1 5. The extremities of the minoraxis of a hypferbola are inthe points (0, ± 3) and the eccentricity is 2. Find the equa-tion of the hyperbola. w,/ - 7^. Jb s r i 6. Show that, in terms of a and h, e has the value -^ ■VaF+¥e = —• a f. Express b in terms of a and e. 8. Prove that two hyperbolas which have the same eccen-tricity are similar, and conversely. 9. Establish formulas (3). y f . H 4. The Asymptotes. Two lines, called the asymptotes, stand in a peculiar and important relation to the hyperbola. Theyare the lines f , «u bx ■ „^-, <■/ 6a;y = — and y = a a \j ^- \ .XT oKl^ 130 ANALYTIC GEOMETRY P:(x,y). Let a point P: (x, y) move offalong a branch of the hyperbola (1) a ^ _ 2/^ _ 1 Fig. 7 and let this take place, for def-initeness, in the tirst slope of the line OP is MP ^yOM X Since the coordinates (a;, y) of P satisfy (1), it follows that h (2) and hence (3) y =- Va;2 — a^,a -=-\l---a; a ^ a? When P recedes indefinitely, x increases without liniit, andthe right-hand side of this equation approaches the liniit we see that the slope of OP approaches that of the line OQ, (4) hy = -x, a -X as its limit, always remaining, however, less than the latterslope, so that P is always below OQ. It seems likely that P will come indefinitely near to thisline ; but this fact does not follow from theforegoing, since P might approach a lineparallel to (4) and lying below it. In thatcase, all that has been said would still be true. That P does, however, actually approach(4) can be shown by proving that the dis-tance PQ approaches 0 as its l
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