If `ax + by =3, bx - ay =4 and x ^(2) + y ^(2) = 1,` then the value of `a ^(2) + b ^(2)` is :

Options

A. 17
B. 16
C. 9
D. 25

Answer

To find the value of \( a^2 + b^2 \) given the equations:

1. \( ax + by = 3 \) (Equation 1)
2. \( bx - ay = 4 \) (Equation 2)
3. \( x^2 + y^2 = 1 \) (Equation 3)

We will follow these steps:

### Step 1: Square both equations
We start by squaring both sides of Equation 1 and Equation 2.

From Equation 1:
\[ (ax + by)^2 = 3^2 \implies a^2x^2 + 2abxy + b^2y^2 = 9 \tag{Equation 4} \]

From Equation 2:
\[ (bx - ay)^2 = 4^2 \implies b^2x^2 - 2abxy + a^2y^2 = 16 \tag{Equation 5} \]

### Step 2: Add the two squared equations
Now, we add Equation 4 and Equation 5:

\[ (a^2x^2 + 2abxy + b^2y^2) + (b^2x^2 - 2abxy + a^2y^2) = 9 + 16 \]

This simplifies to:
\[ (a^2 + b^2)x^2 + (a^2 + b^2)y^2 = 25 \]

### Step 3: Factor out common terms
We can factor out \( a^2 + b^2 \):
\[ (a^2 + b^2)(x^2 + y^2) = 25 \]

### Step 4: Substitute the value of \( x^2 + y^2 \)
From Equation 3, we know that \( x^2 + y^2 = 1 \). Substituting this into the equation gives us:
\[ (a^2 + b^2)(1) = 25 \]

### Step 5: Solve for \( a^2 + b^2 \)
Thus, we find:
\[ a^2 + b^2 = 25 \]

### Final Answer
The value of \( a^2 + b^2 \) is \( \boxed{25} \).

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