Analysis of two circles in circles

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Question

The circle $x^{2} + y^{2} = 4 cuts the circle x^{2} + y^{2} - 2 x - 4 = 0$ at the points A and B. If the circle $x^{2} + y^{2} - 4 x - k = 0$ passes through A and B then the value of $k$ , is

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Solution

The circle $x^{2} + y^{2} = 4$ cuts the circle $x^{2} + y^{2} - 2 x - 4 = 0 at A (0, 2) and B (0, - 2)$
The circle $x^{2} + y^{2} - 4 x - k = 0$ passes through A and B. Therefore
$0 + 4 - 0 - k = 0 \Rightarrow k = 4$

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