A) \[\frac{4}{13}(3\sqrt{3}+1)\]
B) \[\frac{4}{13}(3\sqrt{3}-1)\]
C) \[\frac{1}{26}(3\sqrt{3}-1)\]
D) \[\frac{1}{26}(3\sqrt{3}+1)\]
Correct Answer: B
Solution :
| Line \[y=\sqrt{3}\,x\] |
| and curve \[{{x}^{3}}+{{y}^{3}}+3xy+5{{x}^{2}}+3{{y}^{2}}+4x+5y-1=0\] |
| Solving (1) & (2) then\[\Rightarrow \]\[{{x}^{3}}+3\sqrt{3}\,{{x}^{3}}+3\sqrt{3}\,{{x}^{2}}+5{{x}^{2}}+9{{x}^{2}}+4x+5\sqrt{3}\,x-1=0\] |
| Let roots \[{{x}_{1}},\]\[{{x}_{2}},\]\[{{x}_{3}}\] |
| Then \[{{x}_{1}}\,{{x}_{2}}\,{{x}_{3}}=\] |
| Co-ordinates of A, B, C are \[({{x}_{1}},\,\,\sqrt{3}\,{{x}_{1}}),\]\[({{x}_{2}},\,\,\sqrt{3}\,{{x}_{2}})\] and \[({{x}_{3}},\,\,\sqrt{3}\,{{x}_{3}})\] respectively. |
| then \[OA.\,\,OB.\,\,OC=8\,{{x}_{1}}\,{{x}_{2}}\,{{x}_{3}}\] |
| \[=\frac{8}{3\sqrt{3}+1}=\frac{8\,(3\sqrt{3}-1)}{26}=\frac{4}{13}\,\,(3\sqrt{3}-1)\] |
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