. Plane and solid analytic geometry . ion of the hyperbola the extremities ofwhose minor axis are in the points (± 3, 0) and whose eccen-tricity is 1^. 5. Prove that the sum of the squares of the reciprocals ofthe eccentricities of the two conjugate hyperbolas 9 16 . 9 16 is equal to unity. 6. Prove the theorem of Ex. 5 for the general pair of conju-gate hyperbolas. •V 7. Show that the foci of a pair of conjugate hyperbolaslie on a circle. 9. Parametric Representation. It is possible to construct ahyperbola, given its axes, AA and BB, by a method much like that of Ch. VII, § 10, for the ellips
. Plane and solid analytic geometry . ion of the hyperbola the extremities ofwhose minor axis are in the points (± 3, 0) and whose eccen-tricity is 1^. 5. Prove that the sum of the squares of the reciprocals ofthe eccentricities of the two conjugate hyperbolas 9 16 . 9 16 is equal to unity. 6. Prove the theorem of Ex. 5 for the general pair of conju-gate hyperbolas. •V 7. Show that the foci of a pair of conjugate hyperbolaslie on a circle. 9. Parametric Representation. It is possible to construct ahyperbola, given its axes, AA and BB, by a method much like that of Ch. VII, § 10, for the ellipse KyP- rp THE HYPERBOLA 143 Let the two circles, C and C, and the ray from 0, beas before. At the point Ldraw the tangent to C, andmark the point Q where theray cuts this line. At R drawthe tangent to C and mark thepoint >S where this tangentcuts the axis of x. The locus of the pointP: (x, y), in which the paral-lel to the axis of x throughQ and the parallel to theaxis of y through S intersect, is the hyperbola. drawn iP^). For, OR = a, OL = b^ and X — OS = a sec , y=LQ = Hence X -= sec<^,a ^ = tan <f),b and since sec cf) ~ tan2 </) = !, it follows that X tb = 1.
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