A) \[{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}=0\]
B) \[{{y}_{1}}+{{y}_{2}}+{{y}_{3}}+{{y}_{4}}=2\]
C) \[{{x}_{1}}{{x}_{2}}{{x}_{3}}{{x}_{4}}=2{{c}^{4}}\]
D) \[{{y}_{1}}{{y}_{2}}{{y}_{3}}{{y}_{4}}=2{{c}^{4}}\]
Correct Answer: A
Solution :
| [a] \[({{x}_{i}},{{y}_{i}}),\,\,i=1,\,\,2,\,\,3,\,\,4\] lies on |
| \[xy={{c}^{2}}\Rightarrow {{y}_{i}}=\frac{{{c}^{2}}}{{{x}_{i}}}\] |
| Now the point \[({{x}_{i}},{{y}_{i}})\] lies on |
| \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\Rightarrow {{x}^{2}}_{i}+\frac{{{c}^{4}}}{{{x}_{i}}}={{a}^{2}}\] |
| \[\Rightarrow {{x}^{4}}_{i}-{{a}^{2}}{{x}^{2}}_{i}+{{c}^{4}}=0\] |
| Its roots are \[{{x}_{1}},{{x}_{2}},{{x}_{3}},{{x}_{4}}\therefore \] |
| \[{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}=0\] |
| \[{{x}_{1}}{{x}_{2}}+{{x}_{1}}{{x}_{3}}+{{x}_{1}}{{x}_{4}}+{{x}_{2}}{{x}_{3}}+{{x}_{2}}{{x}_{4}}+{{x}_{3}}{{x}_{4}}={{a}^{2}}\] |
| \[{{x}_{1}}{{x}_{2}}{{x}_{3}}+{{x}_{1}}{{x}_{2}}{{x}_{4}}+{{x}_{1}}{{x}_{3}}{{x}_{4}}+{{x}_{2}}{{x}_{3}}{{x}_{4}}=0\] |
| \[{{x}_{1}}{{x}_{2}}{{x}_{3}}{{x}_{4}}={{c}^{4}}\] Clearly [c] is not correct |
| Now \[{{y}_{1}}{{y}_{2}}{{y}_{3}}{{y}_{4}}=\frac{{{c}^{2}}}{{{x}_{1}}}.\frac{{{c}^{2}}}{{{x}_{2}}}.\frac{{{c}^{2}}}{{{x}_{3}}}.\frac{{{c}^{2}}}{{{x}_{4}}}={{c}^{4}}\] |
| and \[{{y}_{1}}+{{y}_{2}}+{{y}_{3}}+{{y}_{4}}=\frac{{{c}^{2}}(\Sigma {{x}_{1}}{{x}_{2}}{{x}_{3}})}{{{x}_{1}}{{x}_{2}}{{x}_{3}}{{x}_{4}}}=0\] |
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