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Mathematical conic curve of senior two.
This is just a train of thought. If you have any questions, please ask them again:

(1) the relationship between a and c can be known from eccentricity, and b= 1 can be known from the length of short axis, and A =b +c is added to get a=2.

(2) From the equation of straight line MD and elliptic equation, we can find the coordinates of m points, and by the same token, we can find the coordinates of n points, and then we can find the length of MN.

In this way, the equation of straight line MN can be obtained, and then the distance d from point D to straight line MN can be obtained.

Therefore, s can be represented by k.

The value range of k can be obtained by solving inequalities.

Note that the range of k found here should intersect with the range of k when the above equation has a solution.