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Maths-

General

Easy

Question

The area of an ellipse is 8π sq. units dist. between the foci is $4 square root of 3$ then e=

$s i n 30^{\circ}$
$s i n 45^{\circ}$
$s i n 60^{\circ}$
$s i n 75^{\circ}$

hint Hint:

Compare the two different formulas of eccentricity.

The correct answer is: $s i n space 60 to the power of ring operator$

$A r e a space o f space a n space e l l i p s e space equals space pi a b equals 8 pi space s q. space u n i t s therefore a b equals 8 space space space space..... open parentheses 1 close parentheses F o c i equals 2 a e equals 4 square root of 3 therefore a e equals fraction numerator 4 square root of 3 over denominator 2 end fraction equals 2 square root of 3 therefore e equals fraction numerator 2 square root of 3 over denominator space a end fraction space space space..... open parentheses 2 close parentheses e squared equals 1 minus b squared over a squared F r o m space e q n space 1 space a n d space 2 comma open parentheses fraction numerator 2 square root of 3 over denominator space a end fraction close parentheses squared equals 1 minus open parentheses 8 over a squared close parentheses squared open parentheses fraction numerator 2 square root of 3 over denominator space a end fraction close parentheses squared equals 1 minus open parentheses 64 over a to the power of 4 close parentheses open parentheses fraction numerator 12 over denominator space a squared end fraction close parentheses equals open parentheses fraction numerator a to the power of 4 minus 64 over denominator a to the power of 4 end fraction close parentheses 12 a squared equals a to the power of 4 minus 64 a to the power of 4 minus 12 a squared minus 64 equals 0 a to the power of 4 minus 16 a squared plus 4 a squared minus 64 equals 0 a squared equals 16 therefore a equals plus-or-minus 4 e equals fraction numerator 2 square root of 3 over denominator a end fraction equals fraction numerator 2 square root of 3 over denominator 4 end fraction equals fraction numerator square root of 3 over denominator 2 end fraction equals sin 60 degree$

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