Construct ellipse with given foci such that it is tangent to a given circle

Question

Given two points $F_1$ and $F_2$ and a circle centered at $r_0$ with radius $s$, I'd like to construct the ellipse with foci $F_1$ and $F_2$ that is tangent to the given circle. That is the question.

My attempt:

Let $r = [x, y]^T $ be the position vector of a point in the plane. To simplify the analysis, I'll introduce a new coordinate reference with its origin at the center of the ellipse. This is known, because the center of the ellipse is just the midpoint of the two foci. So let

$ C = \dfrac{1}{2} (F_1 + F_2) $

And define the unit vector

$ u_1 = \dfrac{ F_2 - F_1}{\| F_2 - F_1 \| } $

And let $u_2 $ be a unit vector that is perpendicular to $u_1$. Now define the rotation matrix

$ R = [u_1, u_2] $

By letting the $x'$ axis point along $u_1$ and the $y'$ axis point along $u_2$ , then if $p' = (x',y')$ is the local coordinate of a (world) point $p$ with resepct to this new frame, then then two vectors are related by

$ p = C + R p' $

So that

$ p' = R^T (p - C) $

Using these new coordinates, the two foci are now at $F'_1 = (-c, 0)$ and $F'_2 = (c, 0) $. Similarly, the given center of the circle can be computed as $r'_0 = R^T (r_0 - C) $

Now the equation of the ellipse to be found is

$ \dfrac{x^2}{a^2} + \dfrac{y^2}{a^2 - c^2} = 1 $

where $a^2 $ is unknown.

Let $(x_1, y_1)$ be the tangent point with the circle, then $(x_1, y_1)$ satisfies

$ \dfrac{x_1^2}{a^2} + \dfrac{y_1^2}{a^2 - c^2} = 1 \tag{1}$

$ (x_1 - {r'}_{0x} )^2 + (y_1 - {r'}_{0y} )^2 = s^2 \tag{2}$

And in addition to these two, we have that the gradients are along the same direction, which means

$ ( \dfrac{x_1}{a^2} , \dfrac{y_1}{a^2 - c^2} ) = K ( x_1 - r'_{0x} , y_1 - r'_{0y} ) $

So that

$ \dfrac{1}{a^2} \bigg( x_1 ( y_1 - r'_{0y} ) \bigg) - \dfrac{1}{a^2 - c^2} \bigg( y_1 (x_1 - r'_{0x} ) \bigg)= 0 \tag{3}$

Simplifying $(3)$, it becomes,

$ (a^2 - c^2) \bigg( x_1 y_1 - x_1 r'_{0y} \bigg) - a^2 \bigg( x_1 y_1 - y_1 r'_{0x} \bigg) = 0 $

And further into

$ - c^2 x_1 y_1 + (c^2 - a^2) x_1 r'_{0y} + a^2 y_1 r'_{0x} = 0 \tag{4}$

Equation $(1)$ also simplifies to

$ (a^2 - c^2) x_1^2 + a^2 y_1^2 = a^2 (a^2 - c^2) \tag{5}$

And equation $(2)$ when expanded and re-arranged, becomes

$ x_1^2 + y_1^2 - 2 x_1 r'_{0x} - 2 y_1 r'_{0y} + {r'_{0x}}^2 + {r'_{0y}}^2 - s^2 = 0 \tag{6} $

From this equation, it follows that

$ x_1^2 + y_1^2 = 2 x_1 r'_{0x} + 2 y_1 r'_{0y} + s^2 - {r'_{0x}}^2 - {r'_{0y}}^2 $

Substituting this into $(5)$, it becomes,

$ a^2 ( 2 x_1 r'_{0x} + 2 y_1 r'_{0y} + s^2 - {r'_{0x}}^2 - {r'_{0y}}^2 ) - c^2 x_1^2 =a^2 (a^2 - c^2) \tag{5'} $

Finally, let's define $A = a^2$ , then our three equations become

$ x_1^2 + y_1^2 = 2 x_1 r'_{0x} + 2 y_1 r'_{0y} + s^2 - {r'_{0x}}^2 - {r'_{0y}}^2 \tag{7}$

$ A ( 2 x_1 r'_{0x} + 2 y_1 r'_{0y} + s^2 - {r'_{0x}}^2 - {r'_{0y}}^2 ) - c^2 x_1^2 =A (A - c^2) \tag{8} $

$ - c^2 x_1 y_1 + (c^2 - A) x_1 r'_{0y} + A y_1 r'_{0x} = 0 \tag{9}$

Equations $(7),(8), (9)$ are quadratic in the variables $x_1, y_1$ and $A$.

Using Mathematica or Sage Math these three nonlinear equations (equations $(7),(8),(9)$) can be solved and the value(s) of $a$ obtained.

Answer 1

I have the impression you're looking for the single one solution. You do realise there are two solutions, do you?

P.s. I realise very good this should be a comment and not an answer, but there's no way to add an image to a comment.

Answer 2

Comment: Some graphical construction for particular cases, where the ellipse and the circle are not concentric. In all cases we have two solutions.

1- construction $N_1$ shows the large ellipse tends to becom a corc;e when the center of the circle tends to be collinear with major axis of the ellipse.

2- construction $O_1$ shows the case where the center of the circle makes an isosceles triangle with the foci of the ellipse.

3- Construction $P_1$ shows another case.

Answer 3

Using the method developed in question statement, I've worked an example where $F_1 = (2,1), F_2 = (7, 5), r_0 = (5, 10), s = 3 $. There were four solutions in total, 2 ellipses, and 2 hyperbolas. They are shown below.