Let $r_{1}$ and $r_{2}$ be the radii of the largest and smallest circles, respectively, which pass through the point (–4, 1) and having their centers on the circumference of the circle $x^{2} + y^{2} + 2 x + 4 y - 4 = 0 .$ If $\frac{r_{1}}{r_{2}} = a + b \sqrt{2}$ , then $a + b$ is equal to :

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answer is C.

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Detailed Solution

$r_{2} = OP + r r_{1} = OP - r r = \sqrt{9 + 9} r = 3 OP = 3 \sqrt{2} \frac{r_{1}}{r_{2}} = \frac{OP + r}{OP - r} = \frac{3 \sqrt{2} + 3}{3 \sqrt{2} - 3} = \frac{\sqrt{2} + 1}{\sqrt{2} - 1} a + b \sqrt{2} = 3 + 2 \sqrt{2} a + b = 2$

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