Circle A has a center at #(4 ,-8 )# and a radius of #3 #. Circle B has a center at #(-2 ,-2 )# and a radius of #2 #. Do the circles overlap? If not, what is the smallest distance between them?

2 Answers
Jul 6, 2017

The circles do not overlap and the shortest distance is #=3.5#

Explanation:

The distance between the centers is

#O_AO_B=sqrt((-2-(4))^2+(-2-(-8))^2)#

#=sqrt(36+36)#

#=sqrt72=8.5#

The sum of the radii is

#r_A+r_B=3+2=5#

As,

#O_AO_B> (r_A+r_B)#

The circles do not overlap.

The smallest distance is

#d=8.5-5=3.5#

graph{((x-4)^2+(y+8)^2-9)((x+2)^2+(y+2)^2-4)(y+x+4)=0 [-25.84, 25.46, -16.57, 9.1]}

Jul 6, 2017

#"no overlap " d~~3.485#

Explanation:

What we have to do here is #color(blue)"compare " #the distance ( d) between the centres of the circles with 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)|)))#
#(x_1,y_1),(x_2,y_2)" are 2 coordinate points"#

#"the points are " (x_1,y_1)=(4,-8),(x_2,y_2)=(-2,-2)#

#d=sqrt((-2-4)^2+(-2+8)^2)=sqrt72~~8.485#

#"sum of radii "=3+2=5#

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

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

#=8.485-5=3.485#
graph{(y^2+16y+x^2-8x+71)(y^2+4y+x^2+4x+4)=0 [-25.31, 25.32, -12.66, 12.65]}