Circle A has a center at #(-4 ,-3 )# and a radius of #2 #. Circle B has a center at #(2 ,-1 )# and a radius of #3 #. Do the circles overlap? If not what is the smallest distance between them?

Question

Circle A has a center at #(-4   ,-3  )# and a radius of #2 #.  Circle B has a center at #(2   ,-1  )# and a radius of #3 #.  Do the circles overlap?  If not what is the smallest distance between them?

Noah Atkinson · Accepted Answer

no overlap, ≈ 1.325

What we have to do here is compare the distance (d) between the centres of the circles to the 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)(a/a)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(a/a)|)))# where # (x_1,y_1)" and " (x_2,y_2)" are 2 coordinate points"# the 2 points here are (-4 ,-3) and (2 ,-1) the centres of the circles. let # (x_1,y_1)=(-4,-3)" and " (x_2,y_2)=(2,-1)# #d=sqrt((2+4)^2+(-1+3)^2)=sqrt(36+4)=sqrt40≈6.325# sum of radii = radius of A + radius of B = 2 + 3 = 5 Since sum of radii < d, then there is no overlap min. distance between them = d - sum of radii #=6.325 - 5 = 1.325 # graph{(y^2+6y+x^2+8x+21)(y^2+2y+x^2-4x-4)=0 [-10, 10, -5, 5]}

Mason Adams · Answer

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

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the distance formula:

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

For Circle A with center $(-4, -3)$ and Circle B with center $(2, -1)$:

$$ d = \sqrt{(2 - (-4))^2 + (-1 - (-3))^2} $$
$$ d = \sqrt{(6)^2 + (2)^2} $$
$$ d = \sqrt{36 + 4} $$
$$ d = \sqrt{40} $$
$$ d = 2\sqrt{10} $$

The sum of their radii is $2 + 3 = 5$.

Since $2\sqrt{10} > 5$, the circles do not overlap.

To find the smallest distance between them, we subtract the sum of their radii from the distance between their centers:

$$ \text{Smallest distance} = 2\sqrt{10} - 5 $$
$$ \text{Smallest distance} = 2\sqrt{10} - 5 $$

Therefore, the smallest distance between the circles is $ 2\sqrt{10} - 5 $.