If \(z=x+iy\) be such that \(\left|\dfrac{z+\lambda i}{z-\lambda i}\right|=1\), where \(i=\sqrt{-1}\) and \(\lambda\) is a positive real number, then which of the following statements is/are correct? I. \(z\) lies on the line \(y=x\). II. The amplitude of \(z\) is \(\dfrac{\pi}{4}\). Select the answer using the code given below.
Neither I nor II
The condition \(|z+\lambda i| = |z-\lambda i|\) means \(z\) is equidistant from \(\lambda i\) and \(-\lambda i\); expanding with \(z=x+iy\) gives \(4y\lambda=0\), so \(y=0\) (since \(\lambda \neq 0\)). Thus \(z\) lies on the real axis, not on \(y=x\), and its amplitude is \(0\) or \(\pi\), not \(\pi/4\). So neither statement is correct.
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