Let S and S' be two (non - concentric circles with centres Aand B and radii `r_(1),r_(2)and d` be the distance between their centres , then one circle lies completely inside the other circle iff

Options

A. `d=r_(1)+r_(2)`
B. `d=|r_(1)-r_(2)|`
C. `d=|r_(2)-r_(1)|`
D. `dlt|r_(1)-r_(2)|`

Answer

To determine the condition under which one circle lies completely inside another, we can analyze the geometric relationships between the two circles based on their centers and radii.

Let:
- Circle S have center A and radius \( r_1 \).
- Circle S' have center B and radius \( r_2 \).
- The distance between the centers A and B is \( d \).

### Step 1: Understand the scenario where Circle S' lies completely inside Circle S.
For Circle S' to lie completely inside Circle S, the following conditions must be satisfied:
1. The distance between the centers \( d \) must be less than the difference of the radii \( r_1 - r_2 \).
2. This ensures that the farthest point of Circle S' (which is at a distance \( r_2 \) from B) does not reach or exceed the boundary of Circle S.

**Condition:**
\[ d < r_1 - r_2 \]

### Step 2: Understand the scenario where Circle S lies completely inside Circle S'.
For Circle S to lie completely inside Circle S', the following conditions must be satisfied:
1. The distance between the centers \( d \) must be less than the difference of the radii \( r_2 - r_1 \).
2. This ensures that the farthest point of Circle S (which is at a distance \( r_1 \) from A) does not reach or exceed the boundary of Circle S'.

**Condition:**
\[ d < r_2 - r_1 \]

### Step 3: Combine the conditions.
From the above two cases, we can summarize the conditions for one circle to lie completely inside the other as follows:
- For Circle S' to lie inside Circle S: \( d < r_1 - r_2 \)
- For Circle S to lie inside Circle S': \( d < r_2 - r_1 \)

Thus, we can combine these conditions into a single statement:
\[ d < |r_1 - r_2| \]

### Final Answer:
One circle lies completely inside the other circle if:
\[ d < |r_1 - r_2| \]