Let S and S' be the foci of an ellipse and B be any one of the extremities of its minor axis. If S'BS is a right angled triangle with right angle at B and area S'BS = 8 s units, then the length of a latus rectum of the ellipse is

Engineering

Mathematics

Standard Ellipse

Tangent to Ellipse

Question

Let S and S' be the foci of an ellipse and B be any one of the extremities of its minor axis. If $∆$ S'BS is a right angled triangle with right angle at B and area $∆$ S'BS = 8 s units, then the length of a latus rectum of the ellipse is

$2 \sqrt{2}$

$4 \sqrt{2}$

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m_SB · m_S'B = – 1

⇒ $\frac{b}{- ae} \cdot \frac{b}{ae} = - 1 \Rightarrow b^{2} = a^{2} e^{2} \Rightarrow \frac{b^{2}}{a^{2}} = 1 - \frac{b^{2}}{a^{2}} \Rightarrow b^{2} = \frac{a^{2}}{2}$

⇒ $\frac{1}{2}$ (b² + a²e²) = 8 ⇒ b² = 8, a² = 16

$∴$ LLR = $\frac{2 b^{2}}{a} = \frac{2 .8}{4}$ s = 4.