A) \[5{{x}^{2}}+5{{y}^{2}}-60x+7=0\]
B) \[5{{x}^{2}}+5{{y}^{2}}+60x-7=0\]
C) \[5{{x}^{2}}+5{{y}^{2}}-60x-7=0\]
D) \[5{{x}^{2}}+5{{y}^{2}}+60x+7=0\]
E) \[5{{x}^{2}}+5{{y}^{2}}+60x+12=0\]
Correct Answer: D
Solution :
Let the point be\[({{x}_{1}},{{y}_{1}})\] According to question, \[\frac{\sqrt{x_{1}^{2}+y_{1}^{2}+4{{x}_{1}}+3}}{\sqrt{x_{1}^{2}+y_{1}^{2}-6{{x}_{1}}+5}}=\frac{2}{3}\] Squaring both sides, \[\frac{x_{1}^{2}+y_{1}^{2}+4{{x}_{1}}+3}{x_{1}^{2}+y_{1}^{2}-6{{x}_{1}}+5}=\frac{4}{9}\] Þ \[9{{x}_{1}}+9y_{1}^{2}+36{{x}_{1}}+27=4x_{1}^{2}+4y_{1}^{2}-24{{x}_{1}}+20\] Þ \[5x_{1}^{2}+5y_{1}^{2}+60{{x}_{1}}+7=0\] Hence, locus is\[5{{x}^{2}}+5{{y}^{2}}+60x+7=0\].
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