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Question:

If the length of the latus rectum of an ellipse is 4 units and the distance between a focus and its nearest vertex on the major axis is 32 units, then its eccentricity is?

13

12

23

19

Solution:

Focus=(ae,0)and vertex=(a,0)Distance between focus and vertex=a(1−e)=32⟹a󔼨=aeSquaring above equation, we get⟹a2+94𕒷a=a2e2...[1]Length of latus rectum=2b2a=4⟹b2=2a...[2]e2=1−b2a2=1𕒶aa2.. (from [2])=1𕒶a...[3Substituting this in equation[1] we get⟹a2+94𕒷a=a2(1𕒶a)⟹a=94Therefore,e2=1𕒶a=1󔽡=19⟹e=13Hence, answer is option (D)