Circle A has a center at #(2 ,7 )# and an area of #81 pi#. Circle B has a center at #(4 ,3 )# and an area of #36 pi#. Do the circles overlap? If not, what is the shortest distance between them?

Question

Circle A has a center at #(2   ,7  )# and an area of #81 pi#.  Circle B has a center at #(4   ,3   )# and an area of #36 pi#.  Do the circles overlap?  If not, what is the shortest distance between them?

Claire Akin · Accepted Answer

#color(blue)("Circles intersect")#

First we find the radii of A and B.

Area of a circle is #pir^2#

Circle A:

#pir^2=81pi=>r^2=81=>r=9#

Circle B:

#pir^2=36pi=>r^2=36=>r=6#

Now we know the radii of each we can test whether they intersect, touch in one place or do not touch.

If the sum of the radii is equal to the distance between the centres, then the circles touch in one place only.

If the sum of the radii is less than the distance between centres, then the circles do not touch

If the sum of the radii is greater than the distance between centres then the circles intersect.

We find the distance between centres using the distance formula.

#d=sqrt((x_2-x_1)^2+(y_2-y_1^2)#

#d=sqrt((2-4)^2+(7-3)^2)=2sqrt(2)#

Sum of radii:

#9+6=15#

#15>2sqrt(2)#

Circles intersect.

Violet Aldrich · Answer

undefined The circles do not overlap. The shortest distance between their centers can be found using the distance formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Substituting the given coordinates:
Distance = √((4 - 2)^2 + (3 - 7)^2)
Distance = √(2^2 + (-4)^2)
Distance = √(4 + 16)
Distance = √20
Distance ≈ 4.47 units.