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Question Number 88436 by Rio Michael last updated on 10/Apr/20
 show that (for e>1) the equation of a hyperbola with   focus  (±ae,0) and directrix  x = (a/e) is (x^2 /a^2 ) − (y^2 /b^2 )     hence find an equation for the eccencitrity of   the hyperbola
$$\:\mathrm{show}\:\mathrm{that}\:\left(\mathrm{for}\:{e}>\mathrm{1}\right)\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{a}\:\mathrm{hyperbola}\:\mathrm{with}\: \\ $$$$\mathrm{focus}\:\:\left(\pm{ae},\mathrm{0}\right)\:\mathrm{and}\:\mathrm{directrix}\:\:{x}\:=\:\frac{{a}}{{e}}\:\mathrm{is}\:\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }\:−\:\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} } \\ $$$$\:\:\:\mathrm{hence}\:\mathrm{find}\:\mathrm{an}\:\mathrm{equation}\:\mathrm{for}\:\mathrm{the}\:\mathrm{eccencitrity}\:\mathrm{of}\: \\ $$$$\mathrm{the}\:\mathrm{hyperbola} \\ $$

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