Circle A has a center at #(4 ,-8 )# and a radius of #3 #. Circle B has a center at #(-2 ,-2 )# and a radius of #2 #. Do the circles overlap? If not, what is the smallest distance between them?

Question

Circle A has a center at #(4  ,-8   )# and a radius of #3 #.  Circle B has a center at #(-2   ,-2    )# and a radius of #2 #.  Do the circles overlap?  If not, what is the smallest distance between them?

Aurora Ayala · Accepted Answer

The circles do not overlap and the shortest distance is #=3.5#

The distance between the centers is

#O_AO_B=sqrt((-2-(4))^2+(-2-(-8))^2)#

#=sqrt(36+36)#

#=sqrt72=8.5#

The sum of the radii is

#r_A+r_B=3+2=5#

As,

#O_AO_B> (r_A+r_B)#

The circles do not overlap.

The smallest distance is

#d=8.5-5=3.5#

graph{((x-4)^2+(y+8)^2-9)((x+2)^2+(y+2)^2-4)(y+x+4)=0 [-25.84, 25.46, -16.57, 9.1]}

Sophia Alfred · Answer

#"no overlap " d~~3.485# What we have to do here is #color(blue)"compare " #the distance ( d) between the centres of the circles with the #color(blue)" sum of the radii"# #• " if sum of radii " > d" then circles overlap"# #• " if sum of radii "< d" then no overlap"# #"to calculate d use the "color(blue)"distance formula"# #color(red)(bar(ul(|color(white)(2/2)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(2/2)|)))# #(x_1,y_1),(x_2,y_2)" are 2 coordinate points"# #"the points are " (x_1,y_1)=(4,-8),(x_2,y_2)=(-2,-2)# #d=sqrt((-2-4)^2+(-2+8)^2)=sqrt72~~8.485# #"sum of radii "=3+2=5# #"since sum of radii "< d" then no overlap"# #"smallest distance "=d-" sum of radii"# #=8.485-5=3.485# graph{(y^2+16y+x^2-8x+71)(y^2+4y+x^2+4x+4)=0 [-25.31, 25.32, -12.66, 12.65]}

Sophia Alfred · Answer

undefined To determine if the circles overlap, we can calculate the distance between their centers and compare it to the sum of their radii. If the distance between the centers is greater than the sum of the radii, the circles do not overlap. Otherwise, they overlap.

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

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

For Circle A with center (4, -8) and Circle B with center (-2, -2), the distance is:

$$ \sqrt{(-2 - 4)^2 + (-2 - (-8))^2} = \sqrt{(-6)^2 + (6)^2} = \sqrt{36 + 36} = \sqrt{72} \approx 8.49 $$

The sum of the radii of Circle A and Circle B is $3 + 2 = 5$. Since the distance between their centers ($ \sqrt{72} $) is greater than the sum of their radii, the circles do not overlap.

The smallest distance between them is the difference between the distance of their centers and the sum of their radii:

$$ \text{Smallest distance} = \sqrt{72} - 5 \approx 8.49 - 5 = 3.49 $$