20. (12 points) The ellipse $\mathrm { E } : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ has semi-focal distance $c$. The distance from the origin O to the line passing through the points $( c , 0 )$ and $( 0 , b )$ is $\frac { 1 } { 2 } c$. (I) Find the eccentricity of ellipse E; (II) As shown in the figure, AB is a diameter of circle $\mathrm { M } : ( x + 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = \frac { 5 } { 2 }$. If the ellipse E passes through points A and B, find the equation of ellipse E. [Figure]
20. (12 points) The ellipse $\mathrm { E } : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ has semi-focal distance $c$. The distance from the origin O to the line passing through the points $( c , 0 )$ and $( 0 , b )$ is $\frac { 1 } { 2 } c$.\\
(I) Find the eccentricity of ellipse E;\\
(II) As shown in the figure, AB is a diameter of circle $\mathrm { M } : ( x + 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = \frac { 5 } { 2 }$. If the ellipse E passes through points A and B, find the equation of ellipse E.\\
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