Given a circle 𝑘(𝑆; 𝑟 = 4 𝑐𝑚) and a line |𝐴𝐵| = 2 𝑐𝑚. Determine and construct the set of all centers of circles that touch circle 𝑘 and have radius 𝑟 = |𝐴𝐵|

Given a circle 𝑘 with center 𝑆 and radius 𝑟 = 4 cm, and a line segment 𝐴𝐵 with length 2 cm. The circles that touch circle 𝑘 and have radius 2 cm will either be inside or outside of circle 𝑘. For the circles inside 𝑘, their centers will lie on a circle with the same center 𝑆 and radius 4 cm - 2 cm = 2 cm. This is because the distance from 𝑆 to the center of the smaller circle is the difference of the radii. For the circles outside 𝑘, their centers will lie on a circle with the same center 𝑆 and radius 4 cm + 2 cm = 6 cm. This is because the distance from 𝑆 to the center of the larger circle is the sum of the radii. So, the set of all centers of circles that touch circle 𝑘 and have radius 𝑟 = 2 cm will lie on two circles, one inside 𝑘 with radius 2 cm and one outside 𝑘 with radius 6 cm, both having the same center 𝑆 as circle 𝑘. Construction part: To construct these, you would draw two circles with the same center 𝑆, one with radius 2 cm and the other with radius 6 cm. The points on these two circles represent the centers of all possible circles that touch circle 𝑘 and have radius 𝑟 = 2 cm.