Let H:$$x2a2$$−$$y2b2=1$$ , where ab$$0be$$ a hyperbola in the $$xy-plane$$ whose conjugate axis LM subtends an angle of $$60$$° at one of its vertices N, Let the area of the triangle LMN be $$43LIST-I$$ $$LIST-II$$ P. The length of the conjugate axis of H is $$1$$. $$8$$ Q. The eccentricity of H is $$2$$. $$43$$ R. The distance between the foci of H is $$3$$. $$23$$ S. The length of the latus rectum of H is $$4$$. $$4$$

Question

Let H:$$x2a2$$−$$y2b2=1$$  , where a>b>$$0be$$ a hyperbola in the $$xy-plane$$ whose conjugate axis LM subtends an angle of $$60$$°  at one of its vertices N, Let the area of the triangle LMN be $$43LIST-I$$                                                                                                             $$LIST-II$$            P. The length of the conjugate axis of H is                                   $$1$$. $$8$$            Q. The eccentricity of H is                                                              $$2$$. $$43$$            R. The distance between the foci of H is                                      $$3$$. $$23$$             S. The length of the latus rectum of H is                                      $$4$$. $$4$$

Infinity Learn · Accepted Answer

$$tan30$$°$$=ba$$  ⇒$$a=b3$$ Now are of Δ$$LMN=12$$..b.$$b3$$  $$43=3b2$$  ⇒$$b=2$$&$$a=23$$  P. Length of conjugate axis $$=2$$ $$b=4$$   So P→$$4$$  Q. Eccentricity $$e=23$$ So Q→$$3$$ R . distance between foci $$=2ae=2(23)23=8$$ So R→$$1$$ S. Length of latus rectum $$=2b2a=2(2)223=43$$ So S→$$2$$

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