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The equation of the circle passing through the point of intersection of the circle $x^{2} + y^{2} = 4$ and the line $2 x + y = 1$ and having minimum possible radius is

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$5 x^{2} + 5 y^{2} + 4 x + 9 y - 5 = 0$

$5 x^{2} + 5 y^{2} + 9 x + 8 y - 15 = 0$

$5 x^{2} + 5 y^{2} - 4 x - 2 y - 18 = 0$

$5 x^{2} + 5 y^{2} + 18 x + 6 y - 5 = 0$

answer is D.

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Detailed Solution

$\begin{array}{l} S + λL = 0 \\ (x^{2} + y^{2} - 4) + λ (2 x + y - 1) = 0 \end{array}$
Hence $r = \sqrt{λ^{2} + \frac{λ^{2}}{4} + 4 + λ}$
r is min ie $λ = - 2 / 5$
$∴$ Reqd circle is 5x² + 5y² - 4x - 2y - 18 = 0

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