A) a circle with radius\[\frac{8}{3}\]
B) an ellipse with length of minor axis\[\frac{16}{9}\]
C) an ellipse with length of major axis\[\frac{16}{3}\]
D) a circle with diameter\[\frac{10}{3}\]
Correct Answer: A
Solution :
\[2|x+i(y+3)|=|x+i(y-1)|\] \[=2\sqrt{{{x}^{2}}+{{(y+3)}^{2}}}\sqrt{{{x}^{2}}+{{(y-1)}^{2}}}\] \[=4{{x}^{2}}+4{{(y+3)}^{2}}={{x}^{2}}+{{(y-1)}^{2}}\] \[=3{{x}^{2}}={{y}^{2}}-2y+1-4{{y}^{2}}-24y-36\] \[=3{{x}^{2}}+3{{y}^{2}}+26y+35=0\] \[={{x}^{2}}+{{y}^{2}}+\frac{26}{3}y+\frac{35}{3}=0\] \[=r=\sqrt{0+\frac{169}{9}-\frac{35}{3}}\] \[=\sqrt{\frac{64}{9}}=\frac{8}{3}\]
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