Why is the shortest distance from the focus of the ellipse to the end of the minor axis?

Why is the shortest distance from the focus of the ellipse to the end of the minor axis?

How to prove that the sum of the distances from a point on an ellipse to two focal points is equal to 2A

Ellipse formula: x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1 (a > b > 0)
Two focus (- A, 0) (a, 0)
Let (x, y) be a point on an ellipse
Radical [(x + a) ^ 2 + y ^ 2] + radical [(x-a) ^ 2 + y ^ 2] = the sum of the distances from the point on the ellipse to the two focal points. The definition is 2A. We can directly substitute it for verification
The square error is as follows:
（x+a)^2 + y^2 + (x-a)^2 + y^2 +
2 radical [(x ^ 2 - A ^ 2) ^ 2 + y ^ 4 + y ^ 2 × [(x + a) ^ 2 + (x-a) ^ 2]]
= 2x^2 + 2y^2 + 2a^2 +
2 radical [(x ^ 2 - A ^ 2) ^ 2 + y ^ 4 + y ^ 2 × [2x ^ 2 + 2A ^ 2]] = 4A ^ 2
The transfer items are:
2x^2 + 2y^2 - 2a^2 =
2 radical [(x ^ 2 - A ^ 2) ^ 2 + y ^ 4 + y ^ 2 × [2x ^ 2 + 2A ^ 2]]
Square on both sides:
4x^4 + 4y^4 + 4a^4 + 8x^2×y^2 - 8x^2×a^2 - 8y^2×a^2=
4x^4 - 8a^2×x^2 + 4a^4 + 4y^4 + 8y^2×x^2 + 8y^2×a^2
So the sum of distances is 2A

Is the sum of the distances from any point on the ellipse to the two focal points = the length of the major axis = 2A?

Yes, from the principle of ellipse formation