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 \[y=\sqrt{3}x\] | ?.(1) | |
| and curve | ||
| \[{{x}^{3}}+{{y}^{3}}+3xy+5{{x}^{2}}+3{{y}^{2}}+4x+5y-1=0\] | ?. (2) | |
| 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 \[{{x}_{1}}{{x}_{2}}{{x}_{3}}=\frac{1}{3\sqrt{3}+1}\] |
| Co-ordinates of A, B, C are \[({{x}_{1}},\,\,\sqrt{3}\,\,{{x}_{1}}),\] \[({{x}_{2}},\,\,\sqrt{3}\,\,{{x}_{2}})\] and \[({{x}_{3}},\,\,\sqrt{3}\,\,{{x}_{3}})\] respestively. |
| 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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