The circles $x^{2} + y^{2} - 10 x + 16 = 0$ and $x^{2} + y^{2} = r^{2}$ intersect each other at two distinct points, if

r < 2

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r > 8

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2 < r < 8

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2 < r < 9

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Solution

The correct option is B $2 < r < 8$
For $C_{1} : x^{2} + y^{2} \to 10 x + 16 = 0$ ; Centre = (5, 0), radius $= \sqrt{5^{2} + 0 - 16} = 3$
For $C_{2} : x^{2} + y^{2} = r^{2};$ Centre = (0, 0), radius = r
For intersection, $| r_{1} - r_{1} | < C_{1} C_{2}$
$\Rightarrow r - 3 < 5 \Rightarrow r < 8$ ........ (1)
and $r_{1} + r_{2} > C_{1} C_{2}, r + 3 > 5 \Rightarrow r = 2$ ...... (2)
From (1) and (2), $2 < r < 8.$

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