Question

Circle A has a center at `(-4 ,6 )` and a radius of `4`. Circle B has a center at `(1 ,1 )` and a radius of `2`. Do the circles overlap? If not what is the smallest distance between them?

Answer 1

The circles do not overlap.
Smallest distance between them is 4

Explanation:

If you find the distance between centres then directly compare it to the sum of the radii you can determine if they do overlap or not.

Let the line length be `L`
Let the radius `r_1=4`
Let the radius `r_2=2`

Using Pythagoras

Then `L=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

`L=sqrt( (1-(-4))^2+(1-6)^2)`

`L=sqrt(25+25)`

`L = 10`

The sum of the radii is `r_1+r_2=4+2=6`

Thus `r_1+r_2 < L` so the circles do not overlap

Distance between `->L-r_1-r_2 = 4`

Answer 2

no overlap , distance ≈ 1.071

Explanation:

First step is to calculate the distance between the centres using the `color(blue)" distance formula "`

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

where `(x_1,y_1)" and " (x_2,y_2)" are 2 coordinate points "`

let `(x_1,y_1)=(-4,6)" and " (x_2,y_2)=(1,1)`

hence : `d = sqrt((1-(-4))^2 + (1-6)^2) = sqrt(25+25 )= sqrt50 ≈ 7.071`

now : radius of A + radius of B = 4+2 =6

since sum of radii < distance between centres → no overlap

distance between circles = 7.071 - 6 = 1.071