Trigonometry & Inverse Trigonometry Question - None

Question

If x and y are non-zero real numbers satisfying xy(x² − y²) = x² + y², then find the minimum value of x² + y².

Accepted Answer

Step 1: Put $x = r\cos\theta$ and $y = r\sin\theta$

Step 2: Substitute into the given equation: $xy(x^2 - y^2) = x^2 + y^2$

$r^2\cos\theta\sin\theta \cdot r^2(\cos^2\theta - \sin^2\theta) = r^2$

$r^2\sin 2\theta\cos 2\theta = 2$

Step 3: Simplify: $\frac{\sin 4\theta}{4} = \frac{1}{r^2}$

$r^2 = 4\csc^2 4\theta$

Step 4: The minimum value of $\csc^2 4\theta$ is 1, so $r^2_{min} = 4$

∴ Minimum value of $x^2 + y^2 = 4$

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