# tangent between two circles on parallel lines

• May 26th 2010, 02:25 PM
tangent between two circles on parallel lines
Given the two points (0,0) and (xg,yg) [the things in red].

The lower circle's origin is on the x-axis and it is tangent to the y-axis at 180 deg (it's left-most point).

The upper circle's origin is on y=yg and it is tangent to (xg,yg) at 0 deg (it's right-most point).

The two circles must have a tangent point (xt,yt) such that:
0 < xt < xg and 0 < yt < yg
(that is, the tangent point must be within the rectangle in the picture).

Solving for an equation with which to calculate the radii of the two circles and their tangent point [the things in blue].

Thank you so much for your help!
• May 27th 2010, 05:58 AM
Opalg
Quote:

Given the two points (0,0) and (xg,yg) [the things in red].

The lower circle's origin is on the x-axis and it is tangent to the y-axis at 180 deg (it's left-most point).

The upper circle's origin is on y=yg and it is tangent to (xg,yg) at 0 deg (it's right-most point).

The two circles must have a tangent point (xt,yt) such that:
0 < xt < xg and 0 < yt < yg
(that is, the tangent point must be within the rectangle in the picture).

Solving for an equation with which to calculate the radii of the two circles and their tangent point [the things in blue].

You have not provided enough conditions to pin down the location of the blue spot. All that can be deduced from the given information is that the blue spot must lie somewhere on the straight line connecting the two red spots. If the upper circle has radius r and the lower circle has radius R, then $R+r = \frac{x_g^2+y_g^2}{2x_g}$. But that only tells you what the sum of the two radii is, it does not determine the individual values of r and R.

In terms of r, the coordinates of the blue spot are $x_t = x_g\biggl(1-\frac{2x_gr}{x_g^2+y_g^2}\biggr)$ and $y_t = y_g\biggl(1-\frac{2x_gr}{x_g^2+y_g^2}\biggr)$.
• May 27th 2010, 07:02 AM