Circle A has a center at #(1 ,5 )# and a radius of #3 #. Circle B has a center at #(2 ,-3 )# and a radius of #5 #. Do the circles overlap? If not, what is the smallest distance between them?

Question

Circle A has a center at #(1  ,5   )# and a radius of #3    #.  Circle B has a center at #(2  ,-3   )# and a radius of #5 #.  Do the circles overlap?  If not, what is the smallest distance between them?

Sophia Alfred · Accepted Answer

They do not overlap.

To overlap the distance between centres has to be less than the sum of their radii. Let the sum of radii be #r# Let the distance between radii be #d# #"Centre"_"A" -> (x_1,y_1)->(1,5)# #"Centre"_"B"->(x_2,y_2)->(2,-3)# Sum of radii #-> 3+5=8 =r# #color(blue)("Distance between centres")# Using Pythagoras #d^2 = (x_2-x_1)^2+(y_2-y_1)^2# #d=+-sqrt((2-1)^2+(-3-5)^2)# #d=+-sqrt(1^2+(-8)^2)" "=" "+-8.062# to 3 decimal places As we are comparing sizes the #+-# does not have any significance. As #d>s# the do not overlap

Wyatt Anderson · 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.

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

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

Where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the centers of the circles.

For Circle A, center coordinates are $(1, 5)$ and for Circle B, center coordinates are $(2, -3)$.

Plugging in the values:

$$ \text{Distance} = \sqrt{(2 - 1)^2 + (-3 - 5)^2} $$
$$ \text{Distance} = \sqrt{(1)^2 + (-8)^2} $$
$$ \text{Distance} = \sqrt{1 + 64} $$
$$ \text{Distance} = \sqrt{65} $$

The sum of the radii of Circle A and Circle B is $3 + 5 = 8$.

Since the distance between the centers ($\sqrt{65}$) is greater than the sum of the radii (8), 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:

$$ \text{Smallest distance} = \sqrt{65} - 8 $$