Find the locus of the centres of circles tangent to two fixed circles.
From my initial observations, I strongly think that the locus may be part of a hyperbola or some other conic? (because the distances from the centre to the centres of the other two circles must have like some sort of a constant difference?)
What does everyone else think? Is there an elementary way to prove that this is indeed the locus? If this is indeed a hyperbola, is there any relationship with the asymptotes? What is the angle between them?
Thanks for all your responses!
There are multiple cases. Let's first consider the case where the fixed circles and the moving circle are in each other's exteriors.
Let the fixed circles have radii $a$ and $b$ and the changing circle have radius $r$.
If that changing circle is tangent to both fixed circles, the center of the changing circle is at distance $a+r$ from one fixed center and $b+r$ from the other fixed center. That means the difference of the distances of the changing center from the fixed centers is
$$(a+r)-(b+r)=a-b$$
which is a constant. Thus the locus of the changing center is one branch of a hyperbola if $a\ne b$, or the perpendicular bisector of the line segment between the two fixed centers if $a=b$.
If the locus is a hyperbola (as in the usual case $a\ne b$), you can discover other facts about the locus by rotating and translating the situation onto a Cartesian coordinate system. If the distance between the fixed centers is $2d$ then place the origin at the midpoint between the fixed centers and the centers at the points $(d,0)$ and $(-d,0)$. The hyperbola is defined by the difference $a-b$. Now find the equation of this hyperbola in the standard form
$$\frac{x^2}{A^2}-\frac{y^2}{B^2}=1$$
The usual analytic-geometry methods will answer your other questions about the asymptotes. Can you finish from here?
Now let's consider the case where one of the fixed circles is inside the other. The moving circle must be outside the inner circle and inside the outer circle, between the two circles.
If $a$ is the radius of the outer fixed circle and $b$ is the radius of the inside fixed circle, the distances of the moving center are then $a-r$ and $b+r$. We therefore get that the sum of the distances is the constant $a+b$ and the locus of that moving center is thus an ellipse (if the two fixed centers are distinct) or a circle (if the two fixed centers are the same point).
Then clearly there is no asymptote and your other questions are moot.
For completeness, you should also consider the cases where the fixed circles are exterior to each other but the moving circle includes one or both of the fixed circles, the fixed circles intersect twice (once is covered in one of the cases above) or are identical, and perhaps some other weird cases. I'll leave those to you.