February 6th, 2015, 05:15 PM  #1 
Senior Member Joined: Jan 2015 From: USA Posts: 107 Thanks: 2  Circles impossible problem :(
My teacher went nuts! How do I solve this??? The problem is in the attached picture. Thanx! 
February 6th, 2015, 07:33 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,674 Thanks: 2654 Math Focus: Mainly analysis and algebra 
Since they are circles, they intersect at zero, one or two points. We are told that it is two. Therefore, you can pick two values for $k$ ($m$ and $n$, for example) and equate the two expressions on the left hand side of the equations. Equality (for $m \ne n$) will come at precisely two points (independent of $m$ and $n$) which are the solution to our problem. Alternatively, you can do it by inspection. The term that is independent of $x$ and $y$ is $(1+k)$. So the other terms must sum to $(1+k)$ in order to satisfy the equation. There are two simple combinations of $x$ and $y$ that do this. The two points must be equidistant from the centre of the circle. The radius of the circle is minimised when the centre is on the line segment joining the two points. 

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