Circle A has a radius of #2 # and a center at #(3 ,1 )#. Circle B has a radius of #4 # and a center at #(8 ,3 )#. If circle B is translated by ##, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Question

Circle A has a radius of #2 # and a center at #(3 ,1 )#.  Circle B has a radius of #4 # and a center at #(8 ,3 )#.  If circle B is translated by #<-4 ,-1 >#, does it overlap circle A?  If not, what is the minimum distance between points on both circles?

Aaliyah Anderson · Accepted Answer

They don't overlap. The minimum distance is #2-sqrt(2)#.

The new center of circle B is #(4,2)#. The distance between the centers of the two circles is #sqrt((4-3)^2 + (2-1)^2)= sqrt(2)# Circle B is the larger circle and all points on the circle are 4 units from its center. The radius of circle A is 2 and the center of A is #sqrt(2)# units from the center of Circle B so the farthest any point on A can be from the center of B is #2+sqrt(2)#. Since #2+sqrt(2)<4# no point on A overlaps any point on B. The minimum distance between points on A and B is actually the radius of B, which is 4, minus the farthest any point on A can be from the center of B, which is #2+sqrt(2)#: #4-(2+sqrt(2)) = 2-sqrt(2)#

Theodore Abad · Answer

undefined To determine if circle B overlaps circle A after being translated, calculate the distance between the centers of the circles and compare it to the sum of their radii. If the distance between the centers is less than the sum of the radii, the circles overlap. Otherwise, they do not overlap.

The distance between the centers of circle A and circle B can be calculated using the distance formula:

$$ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

Substituting the coordinates of the centers:

$$ \text{Distance} = \sqrt{(8 - 3)^2 + (3 - 1)^2} $$
$$ \text{Distance} = \sqrt{5^2 + 2^2} $$
$$ \text{Distance} = \sqrt{25 + 4} $$
$$ \text{Distance} = \sqrt{29} $$

The sum of the radii of circle A and circle B is $2 + 4 = 6$.

Comparing the distance between the centers to the sum of the radii:

$$ \sqrt{29} > 6 $$

Since the distance between the centers is greater than the sum of the radii, the circles do not overlap.

To find the minimum distance between points on both circles, subtract the sum of their radii from the distance between their centers:

$$ \text{Minimum distance} = \sqrt{29} - 6 $$

$$ \text{Minimum distance} \approx 1.08 $$

Therefore, the minimum distance between points on both circles is approximately 1.08 units.

Circle A has a radius of #2 # and a center at #(3 ,1 )#. Circle B has a radius of #4 # and a center at #(8 ,3 )#. If circle B is translated by #&lt;-4 ,-1 &gt;#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Circle A has a radius of #2 # and a center at #(3 ,1 )#. Circle B has a radius of #4 # and a center at #(8 ,3 )#. If circle B is translated by #<-4 ,-1 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?