P is the point of intersection of circle x^{2}+y^{2}=r^{2} with the yaxis. 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 xaxis. 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
x^{2}+y^{2}=r^{2}
and
(x1)^{2}+y^{2}=1^{2}
To do so, open the brackets in the second one: x^{2}2x+1+y^{2}=1, or, equivalently, x^{2}+y^{2}=2x
Now use
x^{2}+y^{2}=r^{2} from the first equation to conclude that r^{2}=2x, i.e. x=r^{2}/2.
Using
x^{2}+y^{2}=r^{2} 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 xaxis, we put y=0 to get
Finally we have to find the limit of this expression as r shrinks to 0, i.e.
.
