A) \[44\sqrt{5}\]
B) \[-144\sqrt{5}\]
C) \[144\sqrt{5}\]
D) \[-44\sqrt{5}\]
Correct Answer: B
Solution :
\[a=\frac{2-\sqrt{5}}{2+\sqrt{5}}=\frac{2-\sqrt{5}}{2+\sqrt{5}}\times \frac{2-\sqrt{5}}{2-\sqrt{5}}\] \[=\frac{{{(2-\sqrt{5})}^{2}}}{{{2}^{2}}-{{(\sqrt{5})}^{2}}}=\frac{4+5-4\sqrt{5}}{4-5}\] \[=-9+4\sqrt{5}\] Similarly, \[b=-(9+4\sqrt{5})\] \[\therefore \] \[a+b=(-9+4\sqrt{5})\] \[(-9-4\sqrt{5})=-18\] \[a-b=(-9+4\sqrt{5})\] \[-(-9-4\sqrt{5})=8\sqrt{5}\] \[\therefore \] \[{{a}^{2}}-{{b}^{2}}=(a+b)(a-b)\] \[=-18\times 8\sqrt{5}=-144\sqrt{5}\]
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