A) \[\frac{1}{2}\]
B) \[\frac{4}{5}\]
C) \[\frac{1}{\sqrt{5}}\]
D) \[\frac{3}{5}\]
Correct Answer: D
Solution :
Given that,\[2ae=6\]and\[2b=8\] \[\Rightarrow \] \[ae=3\] and \[b=4\] \[\Rightarrow \]\[\frac{ae}{b}=\frac{3}{4}\] Squaring both sides, we have \[\frac{{{a}^{2}}{{e}^{2}}}{{{b}^{2}}}=\frac{9}{16}\] \[\Rightarrow \] \[{{e}^{2}}=\frac{9}{16}\left( \frac{{{b}^{2}}}{{{a}^{2}}} \right)\] We know that, \[\frac{{{b}^{2}}}{{{a}^{2}}}=1-{{e}^{2}}\] \[\Rightarrow \]\[{{e}^{2}}=\frac{9}{16}(1-{{e}^{2}})\] \[\left( \because \frac{{{b}^{2}}}{{{a}^{2}}}=1-{{e}^{2}} \right)\] \[\Rightarrow \]\[\left( \frac{16+9}{9} \right){{e}^{2}}=1\] \[\Rightarrow \]\[e=\frac{3}{5}\]
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