A) \[\frac{3}{4}\]
B) \[\frac{4}{3}\]
C) \[\frac{3}{5}\]
D) \[\frac{5}{3}\]
Correct Answer: B
Solution :
| \[a=\frac{\sqrt{5}+1}{\sqrt{5}-1}=\frac{\sqrt{5}+1}{\sqrt{5}-1}\times \frac{\sqrt{5}+1}{\sqrt{5}+1}\] |
| \[=\frac{{{(\sqrt{5}+1)}^{2}}}{5-1}=\frac{5+1+2\sqrt{5}}{4}=\frac{3+\sqrt{5}}{2}\] |
| \[\therefore \] \[b=\frac{\sqrt{5}-1}{\sqrt{5}+1}=\frac{3-\sqrt{5}}{2}\] |
| \[\therefore \] \[a+b=\frac{3+\sqrt{5}}{2}+\frac{3-\sqrt{5}}{2}=3\] |
| and \[ab=\frac{\sqrt{5}+1}{\sqrt{5}-1}\times \frac{\sqrt{5}-1}{\sqrt{5}+1}=1\] |
| \[\therefore \] Expression\[=\frac{{{a}^{2}}+ab+{{b}^{2}}}{{{a}^{2}}-ab+{{b}^{2}}}\] |
| \[=\frac{{{(a+b)}^{2}}-ab}{{{(a+b)}^{2}}-3ab}\] |
| \[=\frac{9-1}{9-3}=\frac{8}{6}=\frac{4}{3}\] |
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