QQuestion

If x(4 – √3) = y(4 + √3) = 1, then the value of x² + y² = ______

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Accepted Answer

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.