question_answer

Let S and S' be the foci of an ellipse and B be any one of the extremities of its minor axis. If \[\Delta S'BS\]is a right  angled triangle with right angle at B and \[(\Delta S'BS)=8sq.\]units, then the length of a latus rectum of the  ellipse is:

A) 4

B) \[2\sqrt{2}\]

C) \[4\sqrt{2}\]

D) 2

Correct Answer: A

Solution :

\[{{m}_{SB}}.\,{{m}_{S'B}}=-1\]

\[{{b}^{2}}={{a}^{2}}{{e}^{2}}\]

\[\frac{1}{2}S'B.SB=8\]

\[{{a}^{2}}{{e}^{2}}+{{b}^{2}}=16\]

\[{{b}^{2}}={{a}^{2}}(1-{{e}^{2}})\]

Using (i), (ii), (iii), a = 4

\[b=2\sqrt{2}\]

\[e=\frac{1}{\sqrt{2}}\]

\[\therefore \]      \[\ell (L.R.)=\frac{2{{b}^{2}}}{a}=4.\]