Question

How to write a system of equation that satisfies the conditions "a circle and an ellipse that do not intersect?"

Answer 1

`{ (x^2+y^2 = 1), (x^2/4+y^2/9=1) :}`

Explanation:

The equation of a circle with centre `(h, k)` and radius `r` may be written:

`(x-h)^2 + (y-k)^2 = r^2`

The equation of an ellipse with horizontal and vertical axes, centre `(h, k)` and semi axes of lengths `a` and `b` can be written:

`(x-h)^2/a^2+(y-k)^2/b^2 = 1`

Perhaps the simplest example of a non-intersecting system would involve a concentric circle and ellipse with the smaller semi axis larger than the radius of the circle.

So we could write:

`{ (x^2+y^2 = 1), (x^2/4+y^2/9=1) :}`

graph{(x^2+y^2-1)(x^2/4+y^2/9-1) = 0 [-10, 10, -5, 5]}

The largest possible finite number of intersections between a circle and an ellipse is `4`, achievable when the radius of the circle is strictly between the semi-minor axis and semi-major axis lengths:

`{ (x^2+y^2 = 4), (x^2+y^2/9=1) :}`

graph{(x^2+y^2-4)(x^2+y^2/9-1) = 0 [-10, 10, -5, 5]}