A) \[(1+3\text{ }cos\text{ }\theta ,\text{ }\sqrt{3}sin\theta )\]
B) \[(1+3\text{ }cos\theta ,\text{ }5\text{ }sin\theta )\]
C) \[(1+3\text{ }cos\theta ,1+\sqrt{5}sin\theta )\]
D) \[(1+3\text{ }cos\theta ,1+5\text{ }sin\theta )\]
E) \[(1+3cos\theta ,\sqrt{5}sin\theta )\]
Correct Answer: E
Solution :
Given that, foci are (3, 0) and\[(-1,0)\] and\[e=\frac{2}{3}\] \[\therefore \] \[2ae=4\] \[\Rightarrow \] \[2\times a\times \frac{2}{3}=4\Rightarrow a=3\] Also, \[e=\sqrt{1-\frac{{{b}^{2}}}{{{a}^{2}}}}\Rightarrow \frac{4}{9}=1-\frac{{{b}^{2}}}{9}\] \[\Rightarrow \] \[\frac{{{b}^{2}}}{9}=1-\frac{4}{9}=\frac{5}{9}\Rightarrow b=\sqrt{5}\] \[\therefore \]Parametric representation of a point is\[(1+3\cos \theta ,\sqrt{5}\sin \theta ).\]
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