Consider an ellipse and a concentric circle. The circle passes through the foci of the ellipse and intersects the ellipse in four distinct points. The length of major axis of the ellipse is 15 units. If S1 and S2 are the foci of the ellipse and area of triangle PS1S2 is 26 sq. units, then eccentricity of the ellipse is equal to (where P is one of points of intersection of ellipse and circle)

Major Axis 15⇒2a=15 ⇒PS1+PS2=15 (Definition of Ellipse) Ar(ΔPS1S2)=26 ⇒12(PS1)(PS2)=26 ⇒(PS1)(PS2)=52 (S1S2)2=(PS1)2+(PS2)2 =(PS1+PS2)2−2(PS1)(PS2) =121 ⇒S1S2=11 ⇒2ae=11 ⇒2(152)e=11 ⇒e=1115

Consider an ellipse and a concentric circle. The circle passes through the foci of the ellipse and intersects the ellipse in four distinct points. The length of major axis of the ellipse is 15 units. If S1 and S2 are the foci of the ellipse and area of triangle PS1S2 is 26 sq. units, then eccentricity of the ellipse is equal to (where P is one of points of intersection of ellipse and circle)

Major Axis 15⇒2a=15 ⇒PS1+PS2=15 (Definition of Ellipse) Ar(ΔPS1S2)=26 ⇒12(PS1)(PS2)=26 ⇒(PS1)(PS2)=52 (S1S2)2=(PS1)2+(PS2)2 =(PS1+PS2)2−2(PS1)(PS2) =121 ⇒S1S2=11 ⇒2ae=11 ⇒2(152)e=11 ⇒e=1115

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Consider an ellipse and a concentric circle. The circle passes through the foci of the ellipse and intersects the ellipse in four distinct points. The length of major axis of the ellipse is 15 units. If S₁ and S₂ are the foci of the ellipse and area of triangle PS₁S₂ is 26 sq. units, then eccentricity of the ellipse is equal to (where P is one of points of intersection of ellipse and circle)

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$\frac{13}{15}$

$\frac{7}{15}$

$\frac{2}{3}$

$\frac{11}{15}$

answer is D.

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Detailed Solution

$M a j o r A x i s 15 \Rightarrow 2 a = 15 \Rightarrow P S_{1} + P S_{2} = 15 (D e f i n i t i o n o f E l l i p s e) A r (Δ P S_{1} S_{2}) = 26 \Rightarrow \frac{1}{2} (P S_{1}) (P S_{2}) = 26 \Rightarrow (P S_{1}) (P S_{2}) = 52 {(S_{1} S_{2})}^{2} = {(P S_{1})}^{2} + {(P S_{2})}^{2} = {(P S_{1} + P S_{2})}^{2} - 2 (P S_{1}) (P S_{2}) = 121 \Rightarrow S_{1} S_{2} = 11 \Rightarrow 2 a e = 11 \Rightarrow 2 (\frac{15}{2}) e = 11 \Rightarrow e = \frac{11}{15}$