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  • If the normal at an end of a latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity e of the ellipse satisfies: Option: 1 <img alt="e^{4}+2e^{2}-1=0" src="https://learn.careers360.com/lat

If the normal at an end of a latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity e of the ellipse satisfies:
Option: 1 e^{4}+2e^{2}-1=0
Option: 2 e^{2}+e-1=0
Option: 3 e^{4}+e^{2}-1=0  
Option: 4 e^{2}+2e-1=0

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\begin{aligned} &\frac{a^{2} x}{x_{1}}-\frac{b^{2} y}{y_{1}}=a^{2} e^{2}\\ &\frac{a^{2} x}{a e}-\frac{b^{2} y}{b^{2}} \cdot a=a^{2} e^{2}\\ &\frac{a x}{e}-a y=a^{2} e^{2} \Rightarrow \frac{x}{e}-y=a e^{2}\\ &\text { passes through }(0, b)\\ &-b=a e^{2} \Rightarrow b^{2}=a^{2} e^{4}\\ &a^{2}\left(1-e^{2}\right)=a^{2} e^{4} \Rightarrow e^{4}+e^{2}=1 \end{aligned}

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