Question 1206482
.
Given the two circles defined by the equations x^(2)-6x+y^(2)+8y=12 and x^(2)+y^(2)=4y, 
find the algebraic equation of the line connecting their centers.
~~~~~~~~~~~~~~~~~~~


<pre>
To find the centers, apply completing the squares separately to x-terms and y-terms in each equation.


                It can be done MENTALLY.


The center of the 1st circle is the point (3,-4).

The center of the 2nd circle is the point (0,2).


The slope of the line through the centers is  m = {{{(-4-2)/(3-0)}}} = {{{-6/3}}} = -2.


So, an equation of the line can be presented in the form

    y-2 = m*(x-0),

or

    y - 2 = -2x,  or  y = -2x + 2,


or in any other equivalent form.


You can check on your own that the presented equations are satisfied with
the coordinates of the centers, so this straight line goes through these points.
</pre>

Solved.