In an ellipse, if the lines joining a focus to the extremities of the minor axis make an equilateral triangle with the minor axis, the eccentricity of the ellipse is

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Solution

Ellipse be $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1,$ focus (ae, 0), extremities of minor axes (0, b), (0, -b).
$(a e)^{2} + b^{2} = (2 b)^{2} \Rightarrow 3 b^{2} = a^{2} e^{2} \Rightarrow 3 a^{2} (1 - e^{2}) = a^{2} e^{2} \Rightarrow e^{2} = \frac{3}{4} \Rightarrow e = \frac{\sqrt{3}}{2}$

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In an ellipse, if the lines joining a focus to the extremities of the minor axis make an equilateral triangle with the minor axis, the eccentricity of the ellipse is

Ellipse be x2a2+y2b2=1, focus (ae, 0), extremities of minor axes (0, b), (0, -b). (ae)2+b2=(2b)2⇒3b2=a2e2⇒3a2(1−e2)=a2e2⇒e2=34⇒e=√32

Ellipse be $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1,$ focus (ae, 0), extremities of minor axes (0, b), (0, -b).
$(a e)^{2} + b^{2} = (2 b)^{2} \Rightarrow 3 b^{2} = a^{2} e^{2} \Rightarrow 3 a^{2} (1 - e^{2}) = a^{2} e^{2} \Rightarrow e^{2} = \frac{3}{4} \Rightarrow e = \frac{\sqrt{3}}{2}$