My Math Forum Circles impossible problem :(

 Geometry Geometry Math Forum

February 6th, 2015, 05:15 PM   #1
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Joined: Jan 2015
From: USA

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Circles impossible problem :(

My teacher went nuts! How do I solve this???

The problem is in the attached picture. Thanx!
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 February 6th, 2015, 07:33 PM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,689 Thanks: 2669 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. Thanks from matisolla

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