SOLUTION: Hello, I am in a calculus course, and I am having difficulty solving this problem: Consider the circle C1: (x-1)^2+y^2=1, and the circle C2: x^2+y^2=r^2. For small r values, the

Algebra ->  Circles -> SOLUTION: Hello, I am in a calculus course, and I am having difficulty solving this problem: Consider the circle C1: (x-1)^2+y^2=1, and the circle C2: x^2+y^2=r^2. For small r values, the      Log On


   

Question 268667: Hello, I am in a calculus course, and I am having difficulty solving this problem:
Consider the circle C1: (x-1)^2+y^2=1, and the circle C2: x^2+y^2=r^2. For small r values, the circles intersect. Consider the line that goes through the top of C2 and the intersection point in quadrant 1. As r->0, where does the root of the line tend to?
Thank you for the help!
J
Answer by drk(1908) About Me  (Show Source):
You can put this solution on YOUR website!
OUr two equations are
(i) %28x-1%29%5E2+%2B+y%5E2+=+1
(ii) x%5E2+%2B+y%5E2+=+r%5E2
take (i) - (ii) to get
(iii) %28x-1%29%5E2+-+x%5E2+=+1+-+r%5E2
simplify to get
(iv) -2x+%2B+1+=+1+-+r%5E2
solving for x we get
(v) r+%5E2%2F2
now, we can find y as
(vi) y = sqrt%28r%5E2+-+%28r%5E4%2F4%29%29
or simplified to
(vii) y = %28r%2F2%29%2Asqrt%281-r%5E2%29
Now, the y -intercept of C2 is (0,r).
---
Next we create an equation of a line passing through the y intercept of C2 and the crossing point of the two circles as
(viii) y = %28%28-2%2B2%2Asqrt%281-r%5E2%29%29%2Fr%29X+%2B+r
We want the x intercept or the value of x when y = 0.
This is
(ix) x = -r%5E2%2F%28-2%2B2%2Asqrt%281-r%5E2%29%29
The limit of x as r -> 0, is 1
The root of the line tends to 1.