circlescoordinate systemseuclidean-geometry
1) What is the locus of the center of a circle which touches externally two given circles ?
2) What is the locus of the center of a circle which touches externally a given circle and a given straight line(two circles will lie in one side of the given straight line ?
I have no idea how to encounter these problems. Can anyone give me a hint to proceed?
In your figure.
Circle $(A)$ with center $A$, radius $AC=R$
Circle $(B)$ with center $B$, radius $BC=r$
Circle $(D)$ with center $D$, radius $\varphi$
Let $R>r$
$DA-DB=(R+\varphi)-(r+\varphi)=R-r=$constant
That is, the locus of $D$ is the hyperbola with foci $A$ and $B$
The expressions are bit complicated, I'll state some facts of outer and inner Soddy circles without proofs.
Denote the corresponding quantities for outer and inner with upper and lower case respectively.
\begin{align*} \Delta &= \sqrt{abc(a+b+c)} \\ \text{radius:} \quad r_{\pm} &= \frac{abc}{2\Delta \mp (bc+ca+ab)} \\ \text{centre:} \quad S_{\pm} &= \frac {\left( \frac{\Delta}{a} \mp b \mp c \right) A+ \left( \frac{\Delta}{b} \mp c \mp a \right) B+ \left( \frac{\Delta}{c} \mp a \mp b \right) C } {\left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right) \Delta \mp 2(a+b+c)} \end{align*}
You can get the necessary information by substituting $a=1$, $b=2$, $c=3$, $A=(-1,0)$, $B=(2,0)$ and $C=(-1,4)$ as shown in the figure.
Best Answer
1) Supposing the centers are $O_1$ and $O_2$, we need $XO_1-XO_2=r_1-r_2$, which is a constant, so $X$ varies on a hyperbola.
2) In this case, we need that $\delta(X,l)=OX-r$, which is simply a parabola. (The line l is a translated version of the directrix)