If `ax+by=6, bx-ay=2and x^(2)+y^(2)=4` then what is `(a^(2)+b^(2))`?

Options

A. 2
B. 4
C. 5
D. 10

Answer

To solve the problem step by step, we will follow the approach outlined in the video transcript.

### Step 1: Write down the given equations
We are given three equations:
1. \( ax + by = 6 \) (Equation 1)
2. \( bx - ay = 2 \) (Equation 2)
3. \( x^2 + y^2 = 4 \) (Equation 3)

### Step 2: Square both sides of Equation 1 and Equation 2
We will square the first two equations to help us find \( a^2 + b^2 \).

**Squaring Equation 1:**
\[ (ax + by)^2 = 6^2 \]
Expanding the left side:
\[ a^2x^2 + b^2y^2 + 2abxy = 36 \quad \text{(Equation 4)} \]

**Squaring Equation 2:**
\[ (bx - ay)^2 = 2^2 \]
Expanding the left side:
\[ b^2x^2 + a^2y^2 - 2abxy = 4 \quad \text{(Equation 5)} \]

### Step 3: Add Equation 4 and Equation 5
Now, we will add the two equations we obtained from squaring:

\[ (a^2x^2 + b^2y^2 + 2abxy) + (b^2x^2 + a^2y^2 - 2abxy) = 36 + 4 \]

This simplifies to:
\[ (a^2 + b^2)x^2 + (a^2 + b^2)y^2 = 40 \]

### Step 4: Factor out \( (a^2 + b^2) \)
We can factor out \( (a^2 + b^2) \):
\[ (a^2 + b^2)(x^2 + y^2) = 40 \]

### Step 5: Substitute the value of \( x^2 + y^2 \)
From Equation 3, we know that \( x^2 + y^2 = 4 \). Substituting this value into the equation gives:
\[ (a^2 + b^2)(4) = 40 \]

### Step 6: Solve for \( a^2 + b^2 \)
Now, we can solve for \( a^2 + b^2 \):
\[ a^2 + b^2 = \frac{40}{4} = 10 \]

Thus, the value of \( a^2 + b^2 \) is \( 10 \).

### Final Answer
The final answer is:
\[ \boxed{10} \]