If `(a+b):(a-b)=15:1` then the value of `a^(2)-b^(2)` is:

Options

A. 56
B. 15
C. 112
D. 8

Answer

To solve the problem, we start with the given ratio:

\[ (a + b) : (a - b) = 15 : 1 \]

### Step 1: Express the ratio in terms of a variable
We can express \(a + b\) and \(a - b\) in terms of a variable \(k\):
\[ a + b = 15k \quad \text{and} \quad a - b = k \]

### Step 2: Set up equations
Now we have two equations:
1. \(a + b = 15k\)
2. \(a - b = k\)

### Step 3: Solve for \(a\) and \(b\)
To find \(a\) and \(b\), we can add and subtract these two equations.

**Adding the equations:**
\[ (a + b) + (a - b) = 15k + k \]
\[ 2a = 16k \implies a = 8k \]

**Subtracting the second equation from the first:**
\[ (a + b) - (a - b) = 15k - k \]
\[ 2b = 14k \implies b = 7k \]

### Step 4: Calculate \(a^2 - b^2\)
Now we need to find \(a^2 - b^2\). We can use the identity:
\[ a^2 - b^2 = (a + b)(a - b) \]

We already have \(a + b = 15k\) and \(a - b = k\).

Now substituting these values:
\[ a^2 - b^2 = (15k)(k) = 15k^2 \]

### Step 5: Conclusion
Thus, the value of \(a^2 - b^2\) is:
\[ 15k^2 \]

Since \(k\) is a positive constant, \(a^2 - b^2\) is a multiple of 15.

### Final Answer
The value of \(a^2 - b^2\) is \(15k^2\), which is a multiple of 15.

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