Question

If $ax + by = 6$, $bx - ay = 2$ and $x^2 + y^2 = 4$, then the value of $(a^2 + b^2)$ would be:

• A. 10
• B. 2
• C. 4
• D. 5

Hint:

For pairs like $ax+by$ and $bx-ay$, squaring and adding cancels the $xy$ terms, yielding $(a^2+b^2)(x^2+y^2)$ cleanly.

Answer 1

The Correct Option is A

Step 1: Form the complex product \( (a+ib)(x-iy) = (ax+by)+i(bx-ay) = 6+2i \).

Step 2: Take the modulus of both sides: \( |a+ib|\,|x-iy| = |6+2i| = \sqrt{40} \), and note \( |x-iy|=\sqrt{x^2+y^2}=2 \).

Step 3: So \( |a+ib| = \sqrt{40}/2 = \sqrt{10} \), giving \( a^2+b^2=10 \). \[ \boxed{10} \]