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 :
Length of tangent to the circle \[{{x}^{2}}+{{y}^{2}}+4x+3=0\]is \[\sqrt{x_{1}^{2}+y_{1}^{2}+4{{x}_{1}}+3}\] and length of tangent to the circle \[{{x}^{2}}+{{y}^{2}}-6x+5=0\]is\[\sqrt{x_{1}^{2}+y_{1}^{2}-6{{x}_{1}}+5}\]. \[\therefore \]According to question, \[\frac{\sqrt{x_{1}^{2}+y_{1}^{2}+4{{x}_{1}}+3}}{\sqrt{x_{1}^{2}+y_{1}^{2}-6{{x}_{1}}+5}}=\frac{2}{3}\] \[\Rightarrow \]\[9x_{1}^{2}+9y_{1}^{2}+36{{x}_{1}}+27-4x_{1}^{2}-4y_{1}^{2}+\] \[24{{x}_{1}}-20=0\] \[\Rightarrow \]\[5x_{1}^{2}+5y_{1}^{2}+60{{x}_{1}}+7=0\] \[\therefore \]Locus of point is \[5{{x}^{2}}+5{{y}^{2}}+60x+7=0\]
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