A) \[4+\sqrt{3}\]
B) \[\frac{12}{4-\sqrt{3}}\]
C) \[\frac{12}{4+\sqrt{3}}\]
D) \[\frac{-12}{4+\sqrt{3}}\]
Correct Answer: C
Solution :
\[{{x}^{2}}+{{y}^{2}}+xy-3=0\] | ??.(1) |
\[y=\frac{x}{\sqrt{3}}\] | ??..(2) |
\[\because \]centre of ellipse is (0, 0) and (2) also passes through (0, 0) | |
\[\therefore \]chord will be bisect at (0, 0). | |
\[\because \,\,OP=OQ=r\] |
\[\because \] equation PQ is \[\frac{x-0}{\cos \frac{\pi }{6}}=\frac{y-0}{\sin \frac{\pi }{6}}=\pm \,\,r.\] |
\[\therefore \left( \frac{r\sqrt{3}}{2},\frac{r}{2} \right)\]lies on (1) |
\[\Rightarrow \frac{3{{r}^{2}}}{4}+\frac{{{r}^{2}}}{4}+\frac{{{r}^{2}}\sqrt{3}}{4}-3=0\]\[\Rightarrow {{r}^{2}}=\frac{3\times 4}{(4+\sqrt{3})}\]\[\Rightarrow {{r}^{2}}=\frac{12}{4+\sqrt{3}}\] |
\[\therefore \left| \,OP\, \right|\,\,\,\left| \,OQ\, \right|=\frac{12}{4+\sqrt{3}}\] |
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