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Webwork1, problem12

P is the point of intersection of circle x2+y2=r2 with the y-axis. Q is the point of intersection of that circle with the circle of radius 1 centered at point (1,0). Ler (R,0) be the point where line PQ intersects the x-axis. Find the limit of R as r shrinks to 0.

Solution:

First, find the coordinates of P and Q.

Clearly P is the point (0,r).

To find Q we solve simultaneously the two equations
x2+y2=r2
and 
(x-1)2+y2=12

To do so, open the brackets in the second one: x2-2x+1+y2=1, or, equivalently, x2+y2=2x
Now use  x2+y2=r2 from the first equation to conclude that r2=2x, i.e. x=r2/2.

Using  x2+y2=r2 once again we find
  

Thus Q has coordinates

The line passing through points P=(0,r) and has equation
or

To find the point of intersection of this line with the x-axis, we put y=0 to get

Finally we have to find the limit of this expression as r shrinks to 0, i.e. .






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