# Circle A has a center at #(1 ,3 )# and a radius of #1 #. Circle B has a center at #(-2 ,-5 )# and a radius of #2 #. Do the circles overlap? If not, what is the smallest distance between them?

##### 2 Answers

The distance between the centres is given by:

Since this is larger than 3 units, the sum of the radii, the circles do not overlap.

#### Explanation:

The sum of the two radii is

The distance between the centres is given by:

Since this is larger than 3 units, the circles do not overlap, and this is the distance between their centres.

#### Explanation:

What we have to do here is

#color(blue)"compare"# the distance ( d) between the centres of the circles to the#color(blue)"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)(2/2)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(2/2)|)))#

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

#"the points are " (x_1,y_1)=(1,3),(x_2,y_2)=(-2,-5)#

#d=sqrt((-2-1)^2+(5-3)^2)=sqrt(9+64)=sqrt73~~8.544#

#color(blue)"sum of radii "=1+2=3#

#"since sum of radii "< d" then no overlap"#

#"smallest distance "=d-" sum of radii"#

#rArr"smallest distance "=8.544-3=5.544#

graph{(y^2-6y+x^2-2x+9)(y^2+10y+x^2+4x+25)=0 [-12.49, 12.48, -6.24, 6.25]}