If the sum of the areas of two circles with radii `R_1` and `R_2` is equal to the area of a circle of radius R, then

Options

A. `R_1 + R_2 = R`
B. `R_1^2 + R_2^2 = R^2`
C. `R_1+R_2ltR`
D. `R_1^2+R_2^2ltR^2`

Answer

To solve the problem, we need to use the formula for the area of a circle, which is given by:

\[ \text{Area} = \pi r^2 \]

where \( r \) is the radius of the circle.

### Step-by-Step Solution:

1. **Calculate the area of the first circle**:
The area of the first circle with radius \( R_1 \) is:
\[ A_1 = \pi R_1^2 \]

2. **Calculate the area of the second circle**:
The area of the second circle with radius \( R_2 \) is:
\[ A_2 = \pi R_2^2 \]

3. **Sum the areas of the two circles**:
The total area of the two circles is:
\[ A_1 + A_2 = \pi R_1^2 + \pi R_2^2 \]
This can be factored as:
\[ A_1 + A_2 = \pi (R_1^2 + R_2^2) \]

4. **Set the sum equal to the area of the larger circle**:
The area of the larger circle with radius \( R \) is:
\[ A = \pi R^2 \]
According to the problem, we have:
\[ \pi (R_1^2 + R_2^2) = \pi R^2 \]

5. **Cancel out \( \pi \)** (assuming \( \pi \neq 0 \)):
This simplifies to:
\[ R_1^2 + R_2^2 = R^2 \]

### Conclusion:
From the equation \( R_1^2 + R_2^2 = R^2 \), we can analyze the options given in the problem:

- \( R_1 + R_2 = R \) (not necessarily true)
- \( R_1^2 + R_2^2 = R^2 \) (this is true)
- \( R_1 + R_2 < R \) (not necessarily true)
- \( R_1^2 + R_2^2 < R^2 \) (not true)

Thus, the correct option is:
\[ R_1^2 + R_2^2 = R^2 \]