What is the meaning of the difference between the equations of two non-intersecting circles represent?

Question

The difference between the equations of two intersecting circles gives a linear equation which represents the common chord or the common tangent.

But what about two non-intersecting circles?

I experimented with a number of different circles of varying centre and radius but I can't find an answer. The resulting line is perpendicular to the line joining the centres but its position doesn't make much sense to me.

Any ideas?

Answer 1

Difference of two non intersecting circles is a line perpendicular to the line joining the centre of the two circles.

The line always lies between the two circles(near to the smaller one) and touches neither of them(for non intersecting circles).

The distance of the line from mid-point of centres is equal to "difference of squares of radii, divided by twice the distance between centres". It can be written as : $$ d=\frac{|a^2-b^2|}{2L} $$ where

• $d$ the distance of line from mid of centres
• $a$ and $b$ are radii
• $L$ distance between centre of circles