Describe the conditions necessary, where circle A has a larger radius than circle B, for the two circle to have no points of intersection, while circle B is entirely in the interior of circle A.
To determine the conditions for circle A to have a larger radius than circle B, such that the two circles have no points of intersection and circle B is entirely within the interior of circle A, we need to consider the relationship between their radii and the distance between their centers.
Let's assume the following:
- The center of circle A is denoted as point A.
- The center of circle B is denoted as point B.
- The radius of circle A is denoted as rA.
- The radius of circle B is denoted as rB.
In order for circle B to be entirely within the interior of circle A, it means the distance between the centers of the two circles should be less than the difference between their radii, i.e., the distance from A to B should be less than rA - rB.
Mathematically, this can be expressed as:
distance(A, B) < rA - rB
Furthermore, for the circles to have no points of intersection, the distance between their centers should also be greater than the sum of their radii, i.e., the distance from A to B should be greater than rA + rB.
Mathematically, this can be expressed as:
distance(A, B) > rA + rB
Thus, the conditions necessary for circle A to have a larger radius than circle B, with circle B entirely within the interior of circle A and no points of intersection, are:
1. distance(A, B) < rA - rB
2. distance(A, B) > rA + rB
Note that the distance between the centers of the circles is usually calculated using the distance formula:
distance(A, B) = √((xA - xB)^2 + (yA - yB)^2)
where (xA, yA) and (xB, yB) are the coordinates of points A and B, respectively.