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Two circle of radii a and b touch each other externally and are inscribed in the area bounded by \[y=\sqrt{1-{{x}^{2}}}\] and the x-axis Statement-1: If \[b=\frac{1}{2}\] then \[a=\frac{1}{2}\] Statement-2: Distance between the centre of two circles = a + b

A)  Statement-1 and 2 are true and Statement-2 is correct explanation of Statement-1.

B)  Statement-1 and 2 are true and Statement-2 is not correct explanation of Statement-1.

C)   Statement-1 is true, statement-2 is false.

D)  Statement-1 is false, Statement-2 is true.

Correct Answer: D

Solution :

 Let centres of the circles be \[{{C}_{1}}\]and \[{{C}_{2}}\] \[\Rightarrow \]\[{{C}_{1}}\,\text{is}\,(\sqrt{1-2a,}a){{C}_{2}}\,\text{is}\,(\sqrt{1-2b},b)\] \[\Rightarrow \]\[{{C}_{1}}{{C}_{2}}=a+b=a+\frac{1}{2}\] \[\Rightarrow \]\[1-2a+{{\left( a-\frac{1}{2} \right)}^{2}}={{\left( a+\frac{1}{2} \right)}^{2}}\] \[\Rightarrow \]\[a=\frac{1}{4}.\]