Let S and S′ be foci of an ellipse and B be any one of the extremities of its minor axis. If ΔS′BS is a right angled triangle with right angle at B and area of △S′BS=8sq. units, then the length of a latus rectum of the ellipse (in units) is :
A
2
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B
2√2
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C
4√2
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D
4
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Solution
The correct option is D4 Let equation of ellipse be x2a2+y2b2=1,a>b
where S≡(ae,0),S′≡(−ae,0) and B≡(0,b)
∵ΔS′BS is a right angled triangle with right angle at B ∴(S′B)2+(SB)2=(SS′)2⇒a2e2+b2+a2e2+b2=4a2e2⇒b2=a2e2⇒e2=b2a2⇒1−b2a2=b2a2⇒a2=2b2⇒a=√2b
Given, area of △(S′BS)=8sq. units ⇒12×2ae×b=8⇒√2b×b√2b×b=8⇒b2=8∴b=2√2⇒a=4
So, length of latus rectum
=2b2a=2(8)4=4units