Assertion (A) : If circumference of two circles are in the ratio 2 : 3 then ratio area is 4 : 9
Reason (R) : The circumference of a circle is ` 2 pi r ^(2)` and its area is ` pi r ^(2)`

Options

A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true and R is not correct explanation of A
C. A is true but R is false
D. A is false but R is true

Answer

To solve the given question step by step, we will analyze the assertion and reason provided.

### Step 1: Understand the Assertion
The assertion states that if the circumferences of two circles are in the ratio 2:3, then the ratio of their areas is 4:9.

### Step 2: Use the Formula for Circumference
The formula for the circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle.

### Step 3: Set Up the Ratios
Let the circumferences of the two circles be \( C_1 \) and \( C_2 \). According to the assertion:
\[ \frac{C_1}{C_2} = \frac{2}{3} \]
Substituting the circumference formula:
\[ \frac{2\pi r_1}{2\pi r_2} = \frac{2}{3} \]
This simplifies to:
\[ \frac{r_1}{r_2} = \frac{2}{3} \]

### Step 4: Find the Ratio of Areas
The area \( A \) of a circle is given by:
\[ A = \pi r^2 \]
For the two circles, the areas \( A_1 \) and \( A_2 \) can be expressed as:
\[ A_1 = \pi r_1^2 \quad \text{and} \quad A_2 = \pi r_2^2 \]
Now, we can find the ratio of the areas:
\[ \frac{A_1}{A_2} = \frac{\pi r_1^2}{\pi r_2^2} = \frac{r_1^2}{r_2^2} \]

### Step 5: Substitute the Ratio of Radii
From Step 3, we have:
\[ \frac{r_1}{r_2} = \frac{2}{3} \]
Squaring both sides gives:
\[ \left(\frac{r_1}{r_2}\right)^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \]
Thus, we have:
\[ \frac{A_1}{A_2} = \frac{4}{9} \]

### Conclusion
The assertion is true: if the circumferences of two circles are in the ratio 2:3, then the ratio of their areas is indeed 4:9.

### Step 6: Analyze the Reason
The reason states that the circumference of a circle is \( 2\pi r^2 \) and its area is \( \pi r^2 \). However, this is incorrect. The correct formula for the circumference is \( 2\pi r \).

### Final Answer
- Assertion (A) is True.
- Reason (R) is False.