A) \[{{x}^{2}}+{{y}^{2}}+30x-13y-25=0\]
B) \[4{{x}^{2}}+4{{y}^{2}}+30x-13y-25=0\]
C) \[2{{x}^{2}}+2{{y}^{2}}+30x-13y-25=0\]
D) \[{{x}^{2}}+{{y}^{2}}+30x-13y+25=0\]
Correct Answer: B
Solution :
The equation of required circle is\[{{S}_{1}}+\lambda {{S}_{2}}=0\]. Þ \[{{x}^{2}}(1+\lambda )+{{y}^{2}}(1+\lambda )+x(2+13\lambda )-y\left( \frac{7}{2}+3\lambda \right)-\frac{25}{2}=0\] Centre = \[\left( \frac{-(2+13\lambda )}{2},\,\,\frac{\frac{7}{2}+3\lambda }{2} \right)\] \[\because \] Centre lies on \[13x+30y=0\] \[\Rightarrow \]\[-13\left( \frac{2+13\lambda }{2} \right)+30\left( \frac{\frac{7}{2}+3\lambda }{2} \right)=0\]\[\Rightarrow \,\]\[\lambda =1\]. Hence the equation of required circle is \[4{{x}^{2}}+4{{y}^{2}}+30x-13y-25=0.\]
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