Question

Eccentricity of the ellipse whose latus rectum is equal to the distance between two focus points, is

Answer 1

$\textbf{Given:}$

$\textsf{Latusrectum of ellipse is equal to distance between foci}$

$\textbf{To find:}$

$\textsf{Eccentricity of the ellipse}$

$\textbf{Solution:}$

$\textsf{As per given data,}$

$\mathsf{\dfrac{2b^2}{a}=2ae}$

$\implies\mathsf{\dfrac{b^2}{a}=ae}$

$\implies\mathsf{b^2=a^2e}$

$\implies\mathsf{a^2(1-e^2)=a^2e}$

$\implies\mathsf{1-e^2=e}$

$\implies\mathsf{e^2+e-1=0}$

$\textsf{Apply quadratic formula}$

$\mathsf{e=\dfrac{-b\pm\,\sqrt{b^2-4ac}}{2a}}$

$\mathsf{e=\dfrac{-1\pm\,\sqrt{1^2-4{\times}1{\times}(-1)}}{2{\times}1}}$

$\mathsf{e=\dfrac{-1\pm\,\sqrt{1+4}}{2}}$

$\mathsf{e=\dfrac{-1\pm\,\sqrt{5}}{2}}$

$\mathsf{e=\dfrac{-1+\sqrt{5}}{2},\;\dfrac{-1-\sqrt{5}}{2}}$

$\mathsf{But\;e\;>\;0}$

$\implies\boxed{\mathsf{e=\dfrac{\sqrt{5}-1}{2}}}$

$\textbf{Find more:}$

The eccentricity of the conic

$36x^2+144y^2-36x-96y-119=0$

https://brainly.in/question/15964330

The eccentricity of an ellipse with its centre at the origin is 1/2.if one of the directrix is x=4,then equation of ellipse is

https://brainly.in/question/1373399

Answer 2

$\textbf{Given:}$

$\textsf{Latusrectum of ellipse is equal to distance between foci}$

$\textbf{To find:}$

$\textsf{Eccentricity of the ellipse}$

$\textbf{Solution:}$

$\textsf{As per given data,}$

$\mathsf{\dfrac{2b^2}{a}=2ae}$

$\implies\mathsf{\dfrac{b^2}{a}=ae}$

$\implies\mathsf{b^2=a^2e}$

$\implies\mathsf{a^2(1-e^2)=a^2e}$

$\implies\mathsf{1-e^2=e}$

$\implies\mathsf{e^2+e-1=0}$

$\textsf{Apply quadratic formula}$

$\mathsf{e=\dfrac{-b\pm\,\sqrt{b^2-4ac}}{2a}}$

$\mathsf{e=\dfrac{-1\pm\,\sqrt{1^2-4{\times}1{\times}(-1)}}{2{\times}1}}$

$\mathsf{e=\dfrac{-1\pm\,\sqrt{1+4}}{2}}$

$\mathsf{e=\dfrac{-1\pm\,\sqrt{5}}{2}}$

$\mathsf{e=\dfrac{-1+\sqrt{5}}{2},\;\dfrac{-1-\sqrt{5}}{2}}$

$\mathsf{But\;e\;>\;0}$

$\implies\boxed{\mathsf{e=\dfrac{\sqrt{5}-1}{2}}}$