A) \[{{x}^{2}}+{{y}^{2}}-xy+1=0\]
B) \[{{x}^{2}}+{{y}^{2}}-xy\sqrt{3}=1\]
C) \[{{x}^{2}}+{{y}^{2}}=1+xy\sqrt{3}\]
D) \[{{x}^{2}}+{{y}^{2}}-xy\sqrt{3}+1=0\]
Correct Answer: C
Solution :
 |
| \[A\,\,(2\cos \theta ,0)\,\,B\,\,(0,2sin\theta )\,\,C\,\,(h,k)\] |
| From diagram |
| \[h=2\cos \theta +2\cos \,\,(120{}^\circ -\theta )=\cos \theta +\sqrt{3}\sin \theta \] |
| \[k=2\sin (120{}^\circ -\theta )=\sin \theta +\sqrt{3}\cos \theta \] |
| \[{{h}^{3}}+{{k}^{2}}=4+4\sqrt{3}\,\,\sin \theta \cos \theta ;hk\] |
| \[=\sqrt{3}+4\sin \theta \cos \theta \] |
| now eliminating \[\theta \] |
| \[{{h}^{2}}+{{k}^{3}}=\sqrt{3}\,\,hk+1\] |
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