Establish an appropriate plane rectangular coordinate system so that e=( 1, 0).

Let a = (x, y) and b = (m, n).

Ae=x，x= 1，be=m，m=2。

Then a = (1, y) and b = (2, n).

a-b=(- 1，y-n)，ab=2+yn

So 1+(y-n)? =4

Get (y-n)？ =3

So the question becomes (y-n)? =3, find the minimum value of 2+yn

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Solving the third year of high school: inequality method

(y-n)？ =y？ +n？ -2yn

≥2|yn|-2yn，

So 3≥2|yn|-2yn

When yn > 0, it is obviously true.

When yn < 0,3 ≥-4yn, and yn≥-3/4, so 2+yn≥5/4, the minimum value is 5/4.

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Find the solution of senior one, y-n = √ 3, that is, y = n √ 3.

2+yn=2+n？ √3n

The minimum value of the vertex coordinate formula of quadratic function is 5/4.