# Circle A has a center at #(4 ,-1 )# and a radius of #5 #. Circle B has a center at #(-3 ,6 )# and a radius of #2 #. Do the circles overlap? If not, what is the smallest distance between them?

##### 2 Answers

They don't overlap. The closest they get is

#### Explanation:

The radii of the two circles are

We can work out the distance between the centres with Pythagoras. The distance in x is

The closest distance, therefore, will be

#### 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 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 2 points are " (x_1,y_1)=(4,-1),(x_2,y_2)=(-3,6)#

#d=sqrt((-3-4)^2+(6+1)^2)=sqrt(49+49)=sqrt98~~9.899#

#"sum of radii "=5+2=7#

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

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

#color(white)(smallest distance)=9.899-7#

#color(white)(smallest distance)=2.899#

graph{(y^2+2y+x^2-8x-8)(y^2-12y+x^2+6x+41)=0 [-20, 20, -10, 10]}