Circle A has a radius of #3 # and a center of #(2 ,1 )#. Circle B has a radius of #2 # and a center of #(7 ,3 )#. If circle B is translated by #<4 ,2 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
1 Answer
no overlap , ≈ 4.849
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 of circles
Before calculating d, we require to find the 'new' centre of B under the given translation which does not change the shape of the circle only it's position.
Under a translation of
#((4),(2))#
#7,3)to(7+4,3+2)to(11,5)larr" new centre of B"# To calculate d, use the
#color(blue)"distance formula"#
#color(red)(bar(ul(|color(white)(a/a)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(a/a)|)))#
where# (x_1,y_1)" and " (x_2,y_2)" are 2 coordinate points"# The 2 points here are (2 ,1) and (11 ,5)
let
# (x_1,y_1)=(2,1)" and " (x_2,y_2)=(11,5)#
#d=sqrt((11-2)^2+(5-1)^2)=sqrt(81+16)=sqrt97≈9.849# Sum of radii = radius of A + radius of B = 3 + 2 = 5
Since sum of radii < d , then no overlap
min. distance between circles = d - sum of radii
#=9.849-5=4.849#
graph{(y^2-2y+x^2-4x-4)(y^2-10y+x^2-22x+142)=0 [-20, 20, -10, 10]}
