Circle A has a radius of #5 # and a center of #(2 ,7 )#. Circle B has a radius of #1 # and a center of #(6 ,1 )#. 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 #5 # and a center of #(2 ,7 )#.  Circle B has a radius of #1 # and a center of #(6 ,1 )#.  If circle B is translated by #<2 ,7 >#, does it overlap circle A?  If not, what is the minimum distance between points on both circles?

Claire Akin · Accepted Answer

no overlap , min. distance ≈ 0.082 units

What we have to do here is to compare the distance ( d) between the centres of the circles with the sum of the radii. • If sum of radii > d , then circles overlap • If sum of radii < d , then no overlap Before doing this we require to find the new coordinates of the centre of B under the translation, which does not change the shape of the circle, only it's position. Under a translation #((2),(7))# (6 ,1) → (6+2 ,1+7) → (8 ,8) is the new centre of circle B To calculate d, use the #color(blue)"distance formula"# #color(red)(|bar(ul(color(white)(a/a)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(a/a)|)))# where # (x_1,y_1)" and " (x_2,y_2)" are 2 coordinate points"# Here the 2 points are (2 ,7) and (8 ,8) the centres of the circles. let # (x_1,y_1)=(2,7)" and " (x_2,y_2)=(8,8)# #d=sqrt((8-2)^2+(8-7)^2)=sqrt(3 6+1)=sqrt37≈6.082# sum of radii = radius of A + radius of B = 5 + 1 = 6 Since sum of radii < d , then no overlap min. distance between points = d - sum of radii = 6.082 - 6 = 0.082 units (3 decimal paces) graph{(y^2-16y+x^2-16x+127)(y^2-14y+x^2-4x+28)=0 [-52, 52, -26.1, 25.9]}

Noah Atkinson · Answer

undefined To determine if circle B, after being translated by <2, 7>, overlaps circle A, we need to calculate the distance between the centers of the two circles and compare it to the sum of their radii. If the distance between the centers is less than the sum of the radii, then the circles overlap.

First, let's find the distance between the centers of circle A and circle B after the translation:

$$ \text{Distance} = \sqrt{(6 - (2 + 2))^2 + (1 - (7 + 7))^2} $$
$$ = \sqrt{(6 - 4)^2 + (1 - 14)^2} $$
$$ = \sqrt{2^2 + (-13)^2} $$
$$ = \sqrt{4 + 169} $$
$$ = \sqrt{173} $$

The distance between the centers of the two circles after the translation is $ \sqrt{173} $.

Now, let's calculate the sum of their radii:

Sum of radii = Radius of circle A + Radius of circle B
               = 5 + 1
               = 6

The sum of the radii is 6.

Comparing the distance between the centers ($ \sqrt{173} $) and the sum of the radii (6), we can see that $ \sqrt{173} > 6 $. This means that the circles do not overlap after the translation.

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

Minimum distance = Distance between centers - Sum of radii
                 = $ \sqrt{173} - 6 $

Therefore, the minimum distance between points on both circles is $ \sqrt{173} - 6 $.

Circle A has a radius of #5 # and a center of #(2 ,7 )#. Circle B has a radius of #1 # and a center of #(6 ,1 )#. If circle B is translated by #&lt;2 ,7 &gt;#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Circle A has a radius of #5 # and a center of #(2 ,7 )#. Circle B has a radius of #1 # and a center of #(6 ,1 )#. If circle B is translated by #<2 ,7 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?