4

Explanation:

x (4 - \sqrt{3}) = 1 and y (4 + \sqrt{3}) = 1

Solving for x and y:

x = \frac{1}{4 - \sqrt{3}}

y = \frac{1}{4 + \sqrt{3}}

Now, multiplying numerator and denominator by (4 + √3) and (4 – √3) respectively to rationalize:

x = \frac{4 + \sqrt{3}}{13} ,

y = \frac{4 - \sqrt{3}}{13}

Now, squaring both:

x² + y² = \left( \frac{4 + \sqrt{3}}{13} \right)^2 + \left( \frac{4 - \sqrt{3}}{13} \right)^2

Using identity: (a + b)² + (a – b)² = 2(a² + b²)

Solving, we get x² + y² = 4.