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Q.

The set of all real values of for which exactly two common tangents can be drawn to the
circle $x^{2} + y^{2} - 4 x - 4 y + 6 = 0$ and $x^{2} + y^{2} - 10 x - 10 y + λ = 0$ is the interval

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a

(12, 32)

b

(12, 24)

c

(18, 42)

d

(18, 48)

answer is B.

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Detailed Solution

$C_{1} = (2, 2), r_{1} = \sqrt{2} C_{2} = (5, 5), r_{2} = \sqrt{50 - λ} F r o m g i v e n d a t a |r_{1} - r_{2}| < C_{1} C_{2} < r_{1} + r_{2} \Rightarrow |\sqrt{2} - \sqrt{50 - λ}| < \sqrt{18} < \sqrt{2} + \sqrt{50 - λ} \Rightarrow 18 < λ < 42$

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