Let the set of all values of , for which the circles intersect at two distinct points be the interval . Then is equal to [2026]

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Solution

Q.
Let the set of all values of $r$ , for which the circles ${(x + 1)}^{2} + {(y + 4)}^{2} = r^{2} and x^{2} + y^{2} - 4 x - 2 y - 4 = 0$ intersect at two distinct points be the interval $(α, β)$ . Then $α β$ is equal to [2026]

1 24

2 21

3 20

4 25

Ans.
(4)

${(x - 2)}^{2} + {(y - 1)}^{2} = 3^{2} & {(x + 1)}^{2} + {(y + 4)}^{2} = r^{2}$

$| r_{1} - r_{2} |$ $< c_{1} c_{2} < r_{1} + r_{2}$

$| r - 3 |$ $< \sqrt{{(2 + 1)}^{2} + {(1 + 4)}^{2}} < r + 3$

$| r - 3 |$ $< \sqrt{34} & r + 3 > \sqrt{34}$

$- \sqrt{34} < r - 3 < \sqrt{34} & r > \sqrt{34} - 3$

$i.e. r = (3 - \sqrt{34}, 3 + \sqrt{34}) \cap (\sqrt{34} - 3, \infty)$

$i.e. r \in (\sqrt{34} - 3, \sqrt{34} + 3)$

$∴$ $α β = (\sqrt{34} - 3) (\sqrt{34} + 3)$

$= 34 - 9$

$= 25$

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