Let z and w be two complex numbers such that |z|$$=$$|w|$$=1and$$ |$$z+iw$$|$$=$$|z−iw¯|$$=2$$. Then z equals

Question

Let z and w be two complex numbers such that |z|$$=$$|w|$$=1and$$ |$$z+iw$$|$$=$$|z−iw¯|$$=2$$. Then z equals

Infinity Learn · Accepted Answer

We have |−iw|$$=$$|−i||w|$$=1and$$        |iw¯|$$=$$|i|w¯∣$$=1$$⇒−iw and iw¯ lie on the circle |z|$$=1$$. As |z−$$($$−$$iw)$$|$$=$$|z−iw¯|$$=2$$ we get z −iw, as wellas z and i w are the end points of the same diameter, with one end point at z.∴ −$$iw=iw$$¯ ⇒ $$w+w$$¯$$=0$$⇒      w is purely imaginary.Let        $$w=ik$$ where k∈RAs        |w|$$=1$$,we get |ik|$$=1$$⇒ |k|$$=1$$ ⇒ $$k=$$±$$1$$.∴ $$w=$$±i⇒−$$iw=iw$$¯$$=$$±$$1When$$ iw¯$$=1$$,then $$z=$$−$$1$$ andwhen iw¯$$=$$−$$1$$ then $$z=1$$

Let $z$ and $w$ be two complex numbers such that $| z | = | w | = 1$ and $| z + i w | = | z - i \bar{w} | = 2 .$ Then $z$ equals

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Let z and w be two complex numbers such that |z|=|w|=1and |z+iw|=|z−iw¯|=2. Then z equals

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Let $z$ and $w$ be two complex numbers such that $| z | = | w | = 1$ and $| z + i w | = | z - i \bar{w} | = 2 .$ Then $z$ equals