# Circle A has a radius of #5 # and a center of #(2 ,7 )#. Circle B has a radius of #1 # and a center of #(3 ,1 )#. 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?

##### 2 Answers

They do not overlap and the minimum distance between the two circles is >1.

#### Explanation:

To figure this out, the simplest thing to do is to graph it out. From all the information given, we can find the equations for both circles and graph them. A circle's equation is:

Also, the problem gives us the center point at which the circle sits. This will also help with forming the equations.

Circle A is

Circle B is

Now that we have our equations, we can graph them. graph{(x-2)^2+(y-7)^2=25 [-13.39, 14.7, 0.26, 14.31]}

graph{(x-3)^2+(y-1)^2=1 [-11.13, 13.84, -1.94, 10.55]}

Now, let's visualize Circle B translated to the right one, and up three. This will make Circle B's new equation look like:

graph{(x-4)^2+(y-10)^2=1 [-6.25, 13.48, 3.71, 13.58]}

If we now compare the graph of Circle A with the translated graph of Circle B, we can see that they still do not overlap and the minimum distance between the two circles is >1.

#"circle B is inside circle A"#

#### Explanation:

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

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

#• " if difference of radii">d" then 1 circle inside other"#

#"Before calculating d we require to find the centre of B"#

#"under the given translation"#

#"under a translation "<1,3>#

#(3,1)to(3+1,1+3)to(4,4)larrcolor(red)"new centre of B"#

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

#•color(white)(x)d=sqrt((x_2-x_1)^2+(y_2-y_1)^2#

#"let "(x_1,y_1)=2,7)" and "(x_2,y_2)=(4,4)#

#d=sqrt((4-2)^2+(4-7)^2)=sqrt(4+9)=sqrt13~~3.61#

#"sum of radii "=5+1=6#

#"difference of radii "=5-1=4#

#"Since diff. of radii">d" then 1 circle inside other"#

graph{((x-2)^2+(y-7)^2-25)((x-4)^2+(y-4)^2-1)=0 [-40, 40, -20, 20]}