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Tangent Circles

The circles ...

Question

The circles $x^{2} + y^{2} = r^{2}$ and $x^{2} + y^{2} - 10 x + 16 = 0$ intersect in real and distinct points and satisfies the condition $a < r < b$ then $(a, b)$ is given by

A

(2, 10)

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B

(2, 8)

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C

(0, 4)

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D

none of these

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Solution

The correct option is D $(2, 8)$
$C_{1} (0, 0)$ and $r_{1} = r$
$C_{2} (5, 0)$ and $r_{1} = 3$
Since, circle intersect each other
Then, $| r_{1} - r_{2} | < C_{1} C_{2} < | r_{1} + r_{2} |$
$\Rightarrow | r - 3 | < 5 < | r + 3 |$
$\Rightarrow 2 < r < 8$
Ans: B

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