I have two circles that intersect and I need to find the scalar value where they are only touching at one point. Each circle is locked to a point on it's circumference and can only scale relative to this point. Both circles must scale at the same value.
The following has a diagram better visualising the problem. http://tube.geogebra.org/m/1983045
I have been trying to work this problem out, on and off, for a few days without any luck. I'm hoping someone else may have better luck (knowledge) than I with solving this.
What you need is a simple formula: $r_1+r_2=O_1O_2$ where $r_1$ and $r_2$ are radius and $O_1$ and $O_2$ are centers of the circles.
Suppose, the fixed points are $P(x_1,y_1)$ and $Q(x_2,y_2)$ and the slope of the lines the centers are moving on are $s_1,s_2$. Also let the ratio of $\Large {r_2\over r_1}$$=l$
Then let $p=\large{1\over\sqrt{1+{s_1}^2}}$, $q=\large{1\over\sqrt{1+{s_2}^2}}$.
Then $O_1=(x_1+tp,y_1+ts_1p)$ and $O_2=(x_2+ltq,y_2+lts_2q)$ so
$O_1O_2=\sqrt{(x_2+ltq-x_1-tp)^2+(y_2+lts_2q-y_1-ts_1p)^2}$
And we have $r_1=t, r_2=lt$
So you are basically solving
$lt+t=\sqrt{(x_2+ltq-x_1-tp)^2+(y_2+lts_2q-y_1-ts_1p)^2}$ where the only unknown is $t$ and it should not be a complicated equation after you sub in all the numbers.