co-ordinates for centre of known ellipse tangent to known circle

Question

Does any body have the equation for calculating the co-ordinates for the centre of a known ellipse tangent to a known circle

1

Sketch of ellipse and circle attached the 20 degree dimension will be a variable

Answer 1

Let's take the circle centered at $O=(0,0)$, with equation $$\tag{1}x^2+y^2=r^2,$$ where $r=200$, and let $(h,k)$ be the center of the ellipse, whose equation is then $$\tag{2}{(x-h)^2\over a^2}+{(y-k)^2\over b^2}=1,$$ with $a=25$ and $b=40$. We have in addition $$\tag{3}{h\over k}=\tan\alpha$$ (in your example $\alpha=20°$).

If $P=(x,y)$ is the tangency point, the tangent is perpendicular to $OP$ and its slope $y'$ satisfies then $y'= -x/y$, that is: $$ \tag{4}{b^2\over a^2}{x-h\over y-k}={x\over y}. $$ Equations $(1)-(4)$ form a system of degree 8, which can be (numerically) solved for the unknowns $x$, $y$, $h$, $k$. For $\alpha=20°$ I got four real solutions, one of them corresponding to an externally tangent ellipse in the first quadrant: $$ h= 81.53626271442751,\quad k= 224.01904067460944. $$ Another solution corresponds to an internally tangent ellipse, while the other two solutions correspond to reflections of the previous ellipses through point $O$.