Circle

Step 1: The problem involves ellipses with the origin as the focus and as the directrix. For ellipses, the general property is that the sum of distances from any point on the ellipse to the two foci is constant.

Step 2: When we focus on the minor axis of an ellipse, the locus of the end of the minor axis behaves like a parabola. This is because the directrix and the focus define a parabolic shape, a known property of conic sections. The directrix acts as a line, and the focus remains fixed at the origin.

Step 3: Therefore, the locus of the end of the minor axis, given these conditions, forms a parabola.

We are given:

, , , .

The two chords and intersect at right angles at point .

Step 1: Use Geometry of Intersecting Chords.

The formula for the radius of the circle when two chords intersect perpendicularly is:

Step 2: Substitute the Given Values.

Substitute the lengths of , , , and :

Simplify:

Step 3: Calculate the Radius.

Take the square root of both sides:

Conclusion: The radius of the circle is:

1. The equation of a circle passing through and is:

2. Since the circles pass through and :

3. The equation simplifies to:

4. If the circles touch the line orthogonally, the condition for orthogonality is:

Thus, the correct answer is .