What will be the value of xy ,If `x^(2)+ y^(2) = 45 and x - y = 5?`

Options

A. 299
B. 15
C. 20
D. 25

Answer

To find the value of \( xy \) given the equations \( x^2 + y^2 = 45 \) and \( x - y = 5 \), we can follow these steps:

### Step 1: Square the second equation
We start with the equation \( x - y = 5 \). We will square both sides of this equation:
\[ (x - y)^2 = 5^2 \]
This simplifies to:
\[ x^2 - 2xy + y^2 = 25 \]

### Step 2: Substitute \( x^2 + y^2 \)
From the first equation, we know that \( x^2 + y^2 = 45 \). We can substitute this into the equation we obtained from squaring \( x - y \):
\[ x^2 + y^2 - 2xy = 25 \]
Substituting \( x^2 + y^2 = 45 \) gives:
\[ 45 - 2xy = 25 \]

### Step 3: Solve for \( 2xy \)
Now, we can rearrange the equation to isolate \( 2xy \):
\[ 45 - 25 = 2xy \]
This simplifies to:
\[ 20 = 2xy \]

### Step 4: Solve for \( xy \)
Now, divide both sides by 2 to find \( xy \):
\[ xy = \frac{20}{2} = 10 \]

Thus, the value of \( xy \) is \( 10 \).

### Final Answer
\[ \boxed{10} \]