2006 SMT/Geometry Problems/Problem 3

Problem

Circle $\gamma$ is centered at $(0, 3)$ with radius $1$ . Circle $\delta$ is externally tangent to circle $\gamma$ and tangent to the $x$ axis. Find an equation, solved for $y$ if possible, for the locus of possible centers $(x, y)$ of circle $\delta$ .

Solution

$[asy] unitsize(1cm); draw((-5,0)--(5,0)); draw((0,0)--(0,5)); draw(circle((0,3),1)); dot((0,3)); draw(circle((2,3/2),3/2)); dot((2,3/2)); draw((2,3/2)--(2,0)); draw((2,3/2)--(0,3)); label("$\gamma$",(-.5,4),W); label("$\delta$",(7/2,3/2),E); label("$(x,y)$",(2,3/2),ENE); label("$(0,3)$",(0,3),NE); [/asy]$ For $\delta$ to be tangent to both the $x$ axis and $\gamma$ , its distance from $\gamma$ must be equal to its distance from the $x$ axis. Note that its distance from the $x$ axis is just $y$ , and its distance from $\gamma$ is equal its distance from $(0,3)$ minus the radius, which is $1$ . Therefore, by the distance formula,

$\sqrt{(x-0)^2+(y-3)^2}-1=y$ $x^2+y^2-6y+9=(y+1)^2=y^2+2y+1$ $8y=x^2+8$ $\boxed{y=\frac{x^2}{8}+1}$

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2006 SMT/Geometry Problems/Problem 3

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