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If the radical center of the circles $x^{2} + y^{2} = 1$ , $x^{2} + y^{2} - 2 x - 3 = 0$ and $x^{2} + y^{2} - 2 y - 3 = 0$ is $C (α, β)$ and r is the sum of the radii of the given circles, then the circle with $C (α, β)$ as centre and r as radius is

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a

${(x + 1)}^{2} + {(y + 1)}^{2} = 25$

b

${(x - 1)}^{2} + {(y + 1)}^{2} = 4$

c

${(x - 2)}^{2} + {(y - 2)}^{2} = 25$

d

${(x - 1)}^{2} + {(y - 1)}^{2} = 2$

answer is D.

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Detailed Solution

$\begin{array}{l} S : x^{2} + y^{2} = 1 \\ S^{'} : x^{2} + y^{2} - 2 x - 3 = 0 \\ S^{''} : x^{2} + y^{2} - 2 y - 3 = 0 \\ S - S^{'} = 0 \Rightarrow x + 1 = 0 \\ S - S^{''} = 0 \Rightarrow y + 1 = 0 \end{array}$
Radical centre=(-1,-1)
Radius r=1+2+2=5
required circle is ${(x + 1)}^{2} + {(y + 1)}^{2} = 25$

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