A circle of radius R is centered at the origin.

[Icebreaker]

A circle of radius R is centered at the origin. The y-intercept (positive) is at (0, R). As we shift the circle to the left (or right) by n on the x-axis, the y-intercept decreases. Are we able to determine the radius of the circle given a certain proportion between the x-shift and the decrease in y-intercept? Say, if x is shifted by 45, then the y-intercept decreases by 5, what is the radius?

 

[honestrosewater]

I don't know what to do about the degenerative case, but could you solve x² + y² = (x + n)² + (y + m)² given only n and m?
Edit: Maybe I do know what to do in the degenerative case- x = 0. So your example is y² = 45² + (y - 5)²

 

[Icebreaker]

Intuitively, I'm pretty certain that if two circle's radii are different, then the decrease in y-intercept can't be the same, given the same displacement in X.

 

[BicycleTree]

It's easy to see if you draw a diagram.
http://img195.echo.cx/my.php?image=circle9qu.png

So, (y - 5)^2 + 45^2 = y^2 as honestrosewater said.

 

[honestrosewater]

I don't how many solutions there are for y² = 45² + (y - 5)². I was just considering that the circle's center C, the graph's origin O, and the y-intercept Y form a right triangle with line CY as the hypotenuse & radius, and the radius is constant.
Edit: But consider that eventually CY = CO...

 

[Icebreaker]

Yes, I've already worked out (y - 5)^2 + 45^2 = y^2, but I wasn't sure, because a friend did it by drawing out things to scale, and got a different answer. I was afraid that my algebra or the whole concept was wrong.

 

[Kamataat]

I'm tired as hell (3am here), so I may have misread the original problem, but I think a simpler formula for the radius R (w/o having to solve for "y") is:

$$R=\frac{a^2+b^2}{2a}$$

where "a" is the change in y-intercept and "b" is the shift in the x direction.

- Kamataat

 

[BicycleTree]

Kamataat, y is the radius in the formula (y - 5)^2 + 45^2 = y^2. Your formula is simplified, but equivalent.

 

[Kamataat]

Aaaaaaaaargh, I figured something was not right. Anyway, won't make the same mistake of posting when tired again.

- Kamataat

 

[Icebreaker]

Indeed, a general formula for a situation like this may very well be

$$(r+\Delta y)^2 + (\Delta x)^2 = r^2, \Delta x < r/2$$

I'm not sure I can prove it, or whether it has already been proven, or whether it's trivial.