The question is given in the following picture:
The question and its answer is given in the following picture:
I wonder why in the question the author used the word minimal and not minimum, could anyone explain this for me?
Also, the question said the minimal distance between any point on sphere 1 and any point on sphere 2, but the answer speaks about 2 point exactly that are shown in the pink color in the following picture:
Could anyone clarify this for me?
On
Minimum and minimal are I'm sure intended to mean the same thing in this situation. As for your question 'why are they only considering the distance between $2$ specific points?':
Imagine taking $2$ points $x,y$, one on each circle such that the $2$ points are not the exact ones in your picture. Then draw the line connecting them and add in the radii from each point the respective centres of the circles that the points lie on. You should now have a piecewise linear curve connecting the $2$ radii. Since in Euclidean geometry, the shortest curve between $2$ points is a straight line, the length of this curve (which is the sum of the lengths of the radii plus the straight line distance between $x$ and $y$) is strictly greater than the straight line distance between the centres of the circles. Since this distance is the sum of the $2$ radii plus the distance between the $2$ points in your picture we can conclude that the shortest distance between $2$ point on the circles, is attained by the $2$ points in your picture (i.e the points the answer considers).
More mathematically: Let $r_1$, $r_2$ be the lengths of the radii of the circles. Let $x,y$ be points - one on each circle. Let $u,v$ be the points in your picture (i.e the points of intersection of the circles and the straight line between centres). Then by what I described above for all $(x,y) \neq (u,v)$,
$$r_1 + r_2 + \text{distance}(x,y) > r_1 + r_2 + \text{distance}(u,v)$$ so $$\text{distance}(x,y) > \text{distance}(u,v).$$
Hence the minimal distance is attained by the straight line distance between $u$ and $v$.
I think the question meant some thing like this: take any point $(x_1,y_1)$ on first sphere and $(x_2,y_2)$ on second sphere.calculate the distance between these two points. Now by varying $(x_1,y_1)$ on sphere 1 and $(x_2,y_2)$ on sphere 2 we get different distances. The question is to find the pair of points such that this distance is minimum.