A) \[\frac{\sqrt{7}}{2}\]
B) \[\sqrt{\frac{5}{3}}\]
C) \[\sqrt{\frac{3}{2}}\]
D) \[\sqrt{2}\]
Correct Answer: A
Solution :
Equation of chord of contact with respect to point (-4, 2) is \[\frac{-4x}{{{a}^{2}}}\,-\frac{2y}{{{b}^{2}}}=1\] and with respect to point (2, 1) is \[\frac{2x}{{{a}^{2}}}-\frac{y}{{{b}^{2}}}=1\]. Now, according to given condition,\[\left( \frac{\frac{4}{{{a}^{2}}}}{\frac{-2}{{{b}^{2}}}} \right)\,\times \left( \frac{\frac{-2}{{{a}^{2}}}}{\frac{-1}{{{b}^{2}}}} \right)\,=-1\,\Rightarrow \,\frac{{{b}^{4}}}{{{a}^{4}}}=\frac{1}{4}\Rightarrow \,\frac{{{b}^{2}}}{{{a}^{2}}}=\frac{1}{2}\] Now, \[e=\sqrt{1+\frac{{{b}^{2}}}{{{a}^{2}}}}\,=\sqrt{1+\frac{1}{2}}\,=\sqrt{\frac{3}{2}}\]
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