Circle A has a radius of $2$ and a center of $(6 ,5 )$ . Circle B has a radius of $3$ and a center of $(2 ,4 )$ . If circle B is translated by $<1 ,3 >$ , does it overlap circle A? If not, what is the minimum distance between points on both circles?

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Answer 1 · 2017-10-10T18:04:21

Circle B overlaps circle A after translation.

If circle B is translated by < $1,3$ >, the center will be ( $2+1,4+3$ )=( $3,7$ ).
Let circle B' has a radius of $3$ and a center of ( $3,7$ ).

The distance $d$ between the center of circle A and that of circle B' is:
$d=sqrt((3-6)^2+(7-5)^2)=sqrt(13)$

Let $r_a$ and $r_b$ to the radius of circle A and circle B(and B') respectively. $r_a=2, r_b=3$ .

This satisfies the inequation:
$abs(r_a-r_b)< d< r_a+r_b$

Therefore circle A and circle B' (translated circle B) do neither circumscribe nor inscribe. They overlap.
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The figure is cited from http://examist.jp/mathematics/figure-circle/two-circle/ (Japanese)

Circle A has a radius of 2 2 and a center of (6 ,5 )(6 ,5 ). Circle B has a radius of 3 3 and a center of (2 ,4 )(2 ,4 ). If circle B is translated by <1 ,3 ><1 ,3 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?