A) \[\frac{1}{2}\]
B) \[\frac{2}{3}\]
C) \[\frac{1}{9}\]
D) \[\frac{1}{3}\]
Correct Answer: D
Solution :
Focus \[=(ae,0)\]and vertex \[=(a,0)\] Distance between focus and vertex \[=a(1-e)=\frac{3}{2}\] \[\Rightarrow \]\[a-\frac{3}{2}=ae\] Squaring above equation, we get \[\Rightarrow \]\[{{a}^{2}}+\frac{9}{4}-3a={{a}^{2}}{{e}^{2}}\] ?[1] Length of latus rectum \[=\frac{2{{b}^{2}}}{a}=4\] \[\Rightarrow \]\[{{b}^{2}}=2a\] \[{{e}^{2}}=1-\frac{{{b}^{2}}}{{{a}^{2}}}\] ?(from [2]) \[=1-\frac{2}{a}\] ?[3} Substituting this in equation [1] we get \[\Rightarrow \]\[{{a}^{2}}+\frac{9}{4}-3a={{a}^{2}}(1-\frac{2}{a})\] \[\Rightarrow \]\[a=\frac{9}{4}\] Therefore, \[{{e}^{2}}=1-\frac{2}{a}=1-\frac{8}{9}=\frac{1}{9}\] \[\Rightarrow \]\[e=\frac{1}{3}\] Hence, answer is option D
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