Circle A has a radius of #2 # and a center of #(2 ,6 )#. Circle B has a radius of #3 # and a center of #(7 ,8 )#. If circle B is translated by #<-2 ,-3 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

.What we have to do here is #color(blue)"compare "#the distance (d) between the centres to the #color(blue)"sum of the radii"#

#• " if sum of radii"> d" then circles overlap"#

#• " if sum of radii "< d" then no overlap"#

#"Before calculating d we require to find the 'new centre'"# #"of B under the given translation"#

#(7,8)to(7-2,8-3)to(5,5)larrcolor(red)"new centre of B"#

#"to calculate d use the "color(blue)"distance formula"#

#d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#

#"let "(x_1,y_1)=(5,5)" and "(x_2,y_2)=(2,6)#

#d=sqrt((2-5)^2+(6-5)^2)=sqrt(9+1)=sqrt10~~3.16#

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

#"Since sum of radii ">d" then circles overlap"# graph{((x-2)^2+(y-6)^2-4)((x-5)^2+(y-5)^2-9)=0 [-20, 20, -10, 10]}

To determine if circle B overlaps circle A after being translated by <-2, -3>, we need to calculate the distance between the centers of the two circles and compare it to the sum of their radii. The distance between the centers of circles A and B can be found using the distance formula: Distance = √((x2 - x1)^2 + (y2 - y1)^2) For circle A (center at (2, 6)) and circle B (center at (7, 8)): Distance = √((7 - 2)^2 + (8 - 6)^2) = √(5^2 + 2^2) = √(25 + 4) = √29 ≈ 5.39 The sum of the radii of circles A and B is 2 + 3 = 5. Since the distance between the centers of the circles (5.39) is greater than the sum of their radii (5), the circles do not overlap. To find the minimum distance between points on both circles, we subtract the sum of their radii from the distance between their centers: Minimum distance = Distance - (radius_A + radius_B) = 5.39 - (2 + 3) = 5.39 - 5 ≈ 0.39 So, the minimum distance between points on circles A and B is approximately 0.39 units.