Geo5.2.13

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If the length of the latus rectum is equal to half of its minor axis of an ellipse in the standard form,then find the eccentricity of the ellipse.

Let the equation of the ellipse be {\frac  {x^{2}}{a^{2}}}+{\frac  {y^{2}}{b^{2}}}=1\,

Length of the latus rectum is {\frac  {2b^{2}}{a}}=b\,

a=2b\,

The relation between a and b is

b^{2}=a^{2}(1-e^{2})\,

b^{2}=4b^{2}(1-e^{2})\,

e^{2}={\frac  {3}{4}},e={\frac  {{\sqrt  {3}}}{2}}\,

Therefore,the eccentricity of the ellipse is e={\frac  {{\sqrt  {3}}}{2}}\,


Main Page:Geometry:The Ellipse