Circle A has a radius of $3$ $3$ and a center of $(2, 7)$ $(2 ,7 )$ . Circle B has a radius of $2$ $2$ and a center of $(6, 1)$ $(6 ,1 )$ . If circle B is translated by $< 2, 7 >$ $<2 ,7 >$ , does it overlap circle A? If not, what is the minimum distance between points on both circles?

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Answer 1 · 2017-06-28T14:41:44

The circles do not overlap and the minimum distance is $= 1.08$ $=1.08$

Circle $A$ $A$ , center $O_{A} = (2, 7)$ $O_A=(2,7)$

The equation of circle $A$ $A$ is

${(x - 2)}^{2} + {(y - 7)}^{2} = 9$ $(x-2)^2+(y-7)^2=9$

Circle $B$ $B$ , center $O_{B} = (6, 1)$ $O_B=(6,1)$

The equation of circle $B$ $B$ is

${(x - 6)}^{2} + {(y - 1)}^{2} = 4$ $(x-6)^2+(y-1)^2=4$

The center of circle $B'$ after translation is

$(6,1)+(2,7)=(8,8)$

Circle $B'$ , center $O_B'=(8,8)$

The equation of the circle after translation is

$(x-8)^2+(y-8)^2=4$

The distance $O_AO_B'$ is

$=sqrt((8-2)^2+(8-7)^2)$

$=sqrt(36+1)$

$=sqrt37$

$=6.08$

This distance is greater than the sum of the radii

$O_AO_B'>r_A+r_B'$

So, the circles do not overlap and the minimum distance is

$=6.08-(2+3)$

$=1.08$
graph{((x-2)^2+(y-7)^2-9)((x-6)^2+(y-1)^2-4)((x-8)^2+(y-8)^2-4)=0 [-7.28, 18.03, -1.57, 11.09]}

Circle A has a radius of 3 33 and a center of (2 ,7 )(2,7)(2 ,7 ). Circle B has a radius of 2 22 and a center of (6 ,1 )(6,1)(6 ,1 ). If circle B is translated by <2 ,7 ><2,7><2 ,7 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?