The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is:

4/3

$fraction numerator 4 over denominator square root of 3 end fraction$
$fraction numerator 2 over denominator square root of 3 end fraction$
$square root of 3$

Solution

fraction numerator 2 over denominator square root of 3 end fraction

We have, 2b²/a = 8 and 2b = ae
⇒ b² = 4a and 2b = ae
Consider, 2b = ae
Consider, 2b = ae
⇒ 4b² = a²e²
⇒ 4a² (e²-1)=a²e²
⇒ 4e²-4 = e²^{$3 straight e squared space equals space 4 space rightwards double arrow space straight e space equals space fraction numerator 2 over denominator square root of 3 end fraction$ [e>0]}

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