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  4. If the two circles (x-1)2+(y-3)2=r2 and x2+y2-8x+2y+8=0 intersect in two distinct points, then

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Q. If the two circles $(x-1)^2+(y-3)^2=r^2 $ and $ x^2+y^2-8x+2y+8=0 $ intersect in two distinct points, then

AIEEEAIEEE 1989Conic Sections

Solution:

As, the two circles intersect in two distinct points
$\Rightarrow $ Distance between centres lies between $|r_1-r_2| $ and $ |r_1+r_2|.$
i.e. $|r-3| < \sqrt {(4-1)^2+(-1-3)^2} <|r+3| $
$\Rightarrow |r-3|<5<|r+3|$
$\Rightarrow r < 8 $ or $ r>2$
$\therefore 2 < r < 8 $