Circle A has a center at #(5 ,1 )# and an area of #4 pi#. Circle B has a center at #(2 ,8 )# and an area of #9 pi#. Do the circles overlap? If not, what is the shortest distance between them?

Question

Circle A has a center at #(5   ,1  )# and an area of #4 pi#.  Circle B has a center at #(2  ,8   )# and an area of #9 pi#.  Do the circles overlap?  If not, what is the shortest distance between them?

Mason Adams · Accepted Answer

The circles do not overlap
Shortest distance#=d-r_a-r_b=sqrt58-5=2.615# Compute first the distance between the centers #d=sqrt((5-2)^2+(1-8)^2)# #d=sqrt((3^2+(-7)^2)# #d=sqrt((9+49)# #d=sqrt58=7.615# Compute the sum of the radius #r_a+r_b=sqrt4+sqrt9=5# God bless....I hope the explanation is useful.

Autumn Armstrong · Answer

undefined To determine if the circles overlap, we need to check if the distance between their centers is less than the sum of their radii. If the distance between the centers is equal to the sum of their radii, they are tangent to each other. If the distance between the centers is greater than the sum of their radii, they do not overlap.

1. Calculate the distance between the centers of the circles using the distance formula:
   - Let the centers of circles A and B be $A(5, 1)$ and $B(2, 8)$, respectively.
   - Distance $d$ between $A$ and $B$ = $\sqrt{(2 - 5)^2 + (8 - 1)^2}$
   - $d = \sqrt{(-3)^2 + (7)^2}$
   - $d = \sqrt{9 + 49}$
   - $d = \sqrt{58}$

2. Calculate the radii of the circles:
   - Circle A has an area of $4\pi$, so its radius $r_A$ is given by $\sqrt{\frac{4\pi}{\pi}} = 2$.
   - Circle B has an area of $9\pi$, so its radius $r_B$ is given by $\sqrt{\frac{9\pi}{\pi}} = 3$.

3. Check if the circles overlap:
   - If the distance between the centers ($\sqrt{58}$) is greater than the sum of the radii ($2 + 3 = 5$), then the circles do not overlap.

Therefore, the circles do not overlap. The shortest distance between them is the distance between their centers, which is $\sqrt{58}$.