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

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Answer 1 · 2016-04-06T10:05:13

The degree of overlap is

$6-3sqrt(2)~~1.757$ to 3 decimal places

For circle A
Let the centre be $C_a -> (6,2)$
Let radius be $R_a -> 4$

For circle B
Let the centre be $C_b->(5,3)$
Let the radius be $R_b->2$

Let distance between centres be $D$

$C_b$ translated by $<-2,2 >$

$=> C_b ->(5-2,3+2)$

$=> C_b ->(3,5)$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$color(blue)("Determine final distance between centres")$

$D=sqrt((x_2-x_1)^2+(y_2-y_1)^2)$

$D=sqrt((3-6)^2+(5-2)^2)$

$D=sqrt((-3)^2+(3)^2)$

$color(blue)(D=sqrt(18) = 3sqrt(2))$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$color(blue)("Determine if circles overlap")$

For this to be true we need $D < R_a+R_b$

$R_a+R_B = 4+2=6$

$3sqrt(2)<6$ so they do overlap

The degree of overlap is

$6-3sqrt(2)~~1.757$ to 3 decimal places

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