Question

Circle A has a center at `(1 ,4 )` and a radius of `2`. Circle B has a center at `(9 ,3 )` and a radius of `1`. Do the circles overlap? If not what is the smallest distance between them?

Answer 1

`color(green)("They do not overlap.")`
`color(green)("The smallest distance between them is: "sqrt(65)-3"~~5.06`

Explanation:

If the distance between centres is less than the sum of the radii then they overlap.

Let point 1 be `P_1->(x_1,y_1)=(1,4)`
Let point 2 be `P_2->(x_2,y_2)=(9,3)`
Let the distance between centres be `d`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Determine distance between centres")`

`x_("difference")=x_2-x_1 = 9-1=8`
`y_("difference")=y_2=y_1=3-4=-1`

Using Pythagoras `d=sqrt(8^2+(-1)^2) = sqrt(65)`

This is approximately `8`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Determine if they over lap")`

The sum of the radii is `2+1=3`

So if the sum of the radii is 3 and the distance between centres is very close to 8 `color(green)(ul(" they do not overlap."))`

`color(green)("The smallest distance between them is: "sqrt(65)-3"`