A) (12, 32)
B) (18, 42)
C) (12, 24)
D) (18, 48)
Correct Answer: B
Solution :
The equations of the circles are \[{{x}^{2}}+{{y}^{2}}-10x-10y+\lambda =0\] ...(1) and \[{{x}^{2}}+{{y}^{2}}-4x-4y+6=0\] ...(2) \[{{C}_{1}}=\] centre of (1) = (5, 5) \[{{C}_{2}}=\]centre of (2) = (2, 2) d = distance between centres \[={{C}_{1}}{{C}_{2}}=\sqrt{9+9}=\sqrt{18}\] \[{{r}_{1}}=\sqrt{50-\lambda },{{r}_{2}}=\sqrt{2}\] For exactly two common tangents we have \[{{r}_{1}}-{{r}_{2}}<{{C}_{1}}{{C}_{2}}<{{r}_{1}}+{{r}_{2}}\] \[\Rightarrow \]\[\sqrt{50-}\lambda -\sqrt{2}<3\sqrt{2}<\sqrt{50-\lambda }+\sqrt{2}\] \[\Rightarrow \]\[\sqrt{50-}\lambda -\sqrt{2}<3\sqrt{2}\]or\[3\sqrt{2}<3\sqrt{50-\lambda }+\sqrt{2}\] \[\Rightarrow \]\[\sqrt{50-\lambda }<4\sqrt{2}\]or\[2\sqrt{2}<\sqrt{50-\lambda }\] \[\Rightarrow \]\[50-\lambda <32\]or\[\lambda <42\] Required interval is (18, 42)
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