Circles

[bugzpodder]

circle A with radius a is centered at (-a,0)
circle B with radius b is centered at (b-k,0), given 0<k<b<a

express the difference of the non-overlapping regions of circle A and circle B in terms of a,b,and k

[riis]

The difference is the sum of both areas minus twice the overlapping area.
So we need to calculate the overlapping area. The centers of the circles are both on the x axis, so we can suffice to calculate only the upper half of the overlapping area. (The lower half is the upper half mirrored at the x axis).

Since we're only dealing with the upper half, we can express the formulas of both circles in y = f(x).

Circle a: y = sqrt(a - (x + a)²)
Circle b: y = sqrt(b - (x - b + k)²)
Now find the intersection point between a and b. This is point P.
The half overlapping area can be split in two parts by drawing a line from P down to and perpendicular to the x axis. This point is Q = (P.x, 0)
The left half of this area belongs to circle b and the right half to circle a. The area of such a half can be calculated by subtracting a right-angled triangle from a circle part.

For circle A: the corners of the part are: MA, P and O; the corners of the triangle are: MA, P and Q. The area in question is cornered by P, Q and O.

For circle b: the corners of the part are MB, P and (-k, 0); the corners of the triangle are MB, P and Q. The area in question is cornered by P, Q and (-k, 0).

Add both areas, multiply by four and subtract the result from the sum of the areas of both circles.

This should provide you with enough information. Good luck!

[riis]

I'm made a drawing of the location of the mentioned points. See the attachment.

[bugzpodder]

The difference is the sum of both areas minus twice the overlapping area.

I don't think that is true. let me clarify: I want the difference in area of the non-overlapping region in circle A and circle B

[krtxmrtz]

Originally posted by riis
The difference is the sum of both areas minus twice the overlapping area.

I agree with bugzpodder. This would be exactly the sum of both non-overlapping areas. The requested difference is rather, as far as I understand the problem:

Diff. = area of non-ovlp. region of first circle - area of non-ovlp. region of second circle = area of first circle - area of second circle

Or, using this notation:

A: area of first circle
A': area of second circle
A_o: area of overlapping region
A_no: area of non-overlapping region of first circle
A'_no: area of non-overlapping region of second circle

then,

Diff = A_no - A'_no = A - A_o - (A' - A_o) = A - A'

What do you think?

[riis]

Originally posted by bugzpodder
I don't think that is true. let me clarify: I want the difference in area of the non-overlapping region in circle A and circle B

Sorry, I've misinterpreted your question. All the hassle I've mentioned was for nothing, but it was a good exercise for me