# Circle A has a radius of #2 # and a center of #(6 ,3 )#. Circle B has a radius of #3 # and a center of #(2 ,4 )#. If circle B is translated by #<1 ,3 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

##### 1 Answer

circles have one point of contact.

#### 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

#• If sum of radii = d , then one point of contact

Before calculating d, we have to find the ' new' centre of circle B under the given translation, which does not change the shape of the circle only it's position.

#"Under a translation of " ((1),(3))#

#(2,4)to(2+1,4+3)to(3,7)larr" new centre of B"# 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 here are (6 ,3) and (3 ,7)

let

# (x_1,y_1)=(6,3)" and " (x_2,y_2)=(3,7)#

#d=sqrt((3-6)^2+(7-3)^2)=sqrt(9+16)=sqrt25=5# sum of radii = radius of A + radius of B = 2 + 3 = 5

Since sum of radii = d , then circles have one point of contact.

graph{(y^2-6y+x^2-12x+41)(y^2-14y+x^2-6x+49)=0 [-25.64, 25.68, -12.82, 12.83]}