On my own, I am working through a book about learning to do proofs, and I'm stuck with this problem?

Question

Prove that if a, b, and c are real numbers with b <> 0, then the two circles

x^2 + y^2 = a and
(x-b)^2 + y^2 = c

intersect.

I can get x in terms of a, b, and c:

(b^2 + a - c)/2b

and then y^2 = [(b^2 + a - c)/2b] - a,

but when I try multiplying it all out, it doesn't seem to lead anywhere.

Can anyone help? Thanks.

Answer 1

Are you trying to prove that those two circles intersect?
Suppose a = 1, b = 10, c = 2.
Then they don't intersect.

And you're equation for y^2 should be
y^2 = a - [(b^2 + a - c)/2b]^2

If they do intersect, then this will give you the y cordinates of their points of intersection. If they don't intersect, then the RHS will be negative.

Answer 2

Unless I'm misinterpreting what you've said, the statement is false. For example take b = 10, a = 9, c = 9.

x^2 + y^2 = 9 is the circle centered at 0 with radius 3.

(x-10)^2 + y^2 = 9 is the circle centered at 10 with radius 3.

These clearly do not intersect. You can see this easily from the geometry, or you can work it out algebraically as follows:

x^2 + y^2 = 9 = (x-10)^2 + y^2

x^2 = x^2 - 20x + 100

x = -5

So (-5)^2 - 9 = 16 = -y^2.

This is not possible, because y^2 must be nonnegative.

Answer 3

The 1st circle's center is (0,0) and radius sqrt(a).
The 2nd circle's center is (b,0) and radius is sqrt(c).

I assume b<> 0 means b not = 0.

So draw a coordinate system and draw the first circle with center at origin and radius =1 so that a = 1.

Now let b = 10 and draw a circle with center at (10,0) and let c = 1 so the radius is 1. So we have a=1,b=10, and c=1.

These two circles cannot possibly intersect because their centers are separated by a distance of 10 units and their radii are 1 unit. Thus the circles are 8 units apart.

This counterexample shows that your conjecture is false.

Perhaps b <>0 means something else, but to me it says that b is less than 0 and b is >0 so b is any value except 0.

Answer 4

Are you sure this is all of the information? Because if we have a=1, b=100 and c=1, the values are real but the circles won't intersect. You have one unit circle centered at the origin, and a second unit circle with a center way over at (100,0).

Answer 5

the distance between the two centers (0,0) & (b,0) equals b
the radius of the 1st = sqrt a
the radius of the 2nd= sqrt c
the two circles intersect if the distance between the two centers is less than the sum of the two radii i.e b < sqrt a + sqrt c