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)→(3+1,1+3)→(4,4)←new centre of B
to calculate d use the distance formula
∙xd=√(x2−x1)2+(y2−y1)2
let (x1,y1)=2,7) and (x2,y2)=(4,4)
d=√(4−2)2+(4−7)2=√4+9=√13≈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]}
