The area bounded by curve `|x/9|+|y/9|=log_b axxlog_a b` where `a,b > 0, a !=b != 1` is

Options

A. (a)81
B. (b)27
C. (c)162
D. (d)36

Answer

To find the area bounded by the curve \( |x/9| + |y/9| = \log_b (a) \cdot \log_a (b) \), we can follow these steps:

### Step 1: Rewrite the equation
The given equation can be rewritten as:
\[ \frac{|x|}{9} + \frac{|y|}{9} = \log_b (a) \cdot \log_a (b) \]
Multiplying both sides by 9 gives:
\[ |x| + |y| = 9 \cdot \log_b (a) \cdot \log_a (b) \]

### Step 2: Analyze the equation
The equation \( |x| + |y| = k \) (where \( k = 9 \cdot \log_b (a) \cdot \log_a (b) \)) represents a diamond (or rhombus) shape centered at the origin with vertices at \( (k, 0) \), \( (0, k) \), \( (-k, 0) \), and \( (0, -k) \).

### Step 3: Determine the vertices
The vertices of the diamond are:
- \( (9 \cdot \log_b (a), 0) \)
- \( (0, 9 \cdot \log_b (a)) \)
- \( (-9 \cdot \log_b (a), 0) \)
- \( (0, -9 \cdot \log_b (a)) \)

### Step 4: Calculate the area of the diamond
The area \( A \) of a diamond (or rhombus) can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
where \( d_1 \) and \( d_2 \) are the lengths of the diagonals. In this case, both diagonals are equal to \( 2k \):
- \( d_1 = 2 \cdot 9 \cdot \log_b (a) \)
- \( d_2 = 2 \cdot 9 \cdot \log_a (b) \)

Thus, the area becomes:
\[ A = \frac{1}{2} \times (2 \cdot 9 \cdot \log_b (a)) \times (2 \cdot 9 \cdot \log_a (b)) \]
\[ A = \frac{1}{2} \times 36 \cdot \log_b (a) \cdot \log_a (b) \]
\[ A = 18 \cdot \log_b (a) \cdot \log_a (b) \]

### Step 5: Simplify using properties of logarithms
Using the change of base formula, we know:
\[ \log_b (a) = \frac{\log (a)}{\log (b)} \quad \text{and} \quad \log_a (b) = \frac{\log (b)}{\log (a)} \]
Thus,
\[ \log_b (a) \cdot \log_a (b) = 1 \]
So, the area simplifies to:
\[ A = 18 \]

### Conclusion
The area bounded by the curve is \( 18 \).