Circle A has a center at #(3 ,1 )# and an area of #15 pi#. Circle B has a center at #(5 ,2 )# and an area of #24 pi#. Do the circles overlap?

Question

Circle A has a center at #(3   ,1  )# and an area of #15 pi#.  Circle B has a center at #(5  ,2   )# and an area of #24 pi#.  Do the circles overlap?

Evelyn Arboleda · Accepted Answer

Circles intersect each other and greatest possible distance between a point on one circle and another point on the other is #10.908#.

Kindly refer to the information below. The center of first circle is #(3,1)# and as area is #15pi#, the radius is #sqrt15=3.873# (as #pir^2=15pi#, #r=sqrt15#) and center of second circle is #(5,2)# and radius is #sqrt24=4.899#. The distance between centers is #sqrt((5-3)^2+(2-1)^2)# = #sqrt(4+1)=sqrt5=2.236# Let the radii of two circles is #r_1# and #r_2# and we also assume that #r_1>r_2# and the distance between centers is #d#. So here #r_1+r_2=8.772 > d=2.236# and #r_1-r_2=1.026 < d=2.236# Hence, the two circles intersect each other and greatest possible distance between a point on one circle and another point on the other is #3.873+4.899+2.236=10.908# graph{(x^2+y^2-10x-4y+5)(x^2+y^2-6x-2y-5)=0 [-6.25, 13.75, -2.92, 7.08]}

Violet Aldrich · Answer

undefined To determine if two circles overlap, we need to compare the distance between their centers to the sum of their radii.

For Circle A with center (3, 1) and area $15\pi$, we can find its radius using the formula for the area of a circle ($A = \pi r^2$). Therefore, $15\pi = \pi r^2$, which gives us $r^2 = 15$ and $r = \sqrt{15}$.

For Circle B with center (5, 2) and area $24\pi$, we can find its radius in the same way. Thus, $24\pi = \pi r^2$, leading to $r^2 = 24$ and $r = \sqrt{24}$.

The distance between the centers of the circles can be found using the distance formula: 
$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

For Circle A, the center is (3, 1), and for Circle B, the center is (5, 2). Substituting these values into the distance formula, we find: 
$$d = \sqrt{(5 - 3)^2 + (2 - 1)^2} = \sqrt{2^2 + 1^2} = \sqrt{5}$$

Now, we compare the distance between the centers ($\sqrt{5}$) to the sum of their radii ($\sqrt{15} + \sqrt{24}$). If the distance between the centers is greater than the sum of the radii, the circles do not overlap. Otherwise, they do overlap.

Since $\sqrt{5} < \sqrt{15} + \sqrt{24}$, the circles do overlap.