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For a complex number z, let Re(z) denote the real part of z. Let S be the set of all complex numbers z satisfying z4−|z|4 = 4iz2, where i = √-1. Then the minimum possible value of |z1 - z2|2, where z1, z2 ∈ S with Re(z1) > 0 and Re(z2) < 0, is _____
Correct answer is '8'. Can you explain this answer?
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Verified Answer
For a complex number z, let Re(z) denote the real part of z. Let Sbe t...
Let z = x + iy
z4 – |z|4 = 4iz2
⇒ 4ixy = 4i
⇒ xy = 1
z4 – |z|4 = 4iz2
⇒ 4ixy = 4i
⇒ xy = 1
Most Upvoted Answer
For a complex number z, let Re(z) denote the real part of z. Let Sbe t...
Note that $z^4 = (z^2)^2 \geq 0$ for any complex number $z$, so $Re(z^4) \geq 0$. Therefore, we must have $Re(z^4) = 0$ if $z$ is in $S$, which means $z^4$ is purely imaginary.
Let $z = a+bi$ be a complex number in $S$. Then we have:
\begin{align*}
z^4 &= (a+bi)^4 \\
&= a^4 + 4a^3bi - 6a^2b^2 - 4ab^3i + b^4i^4 \\
&= (a^4 - 6a^2b^2 + b^4) + (4a^3b - 4ab^3)i
\end{align*}
Since $z^4$ is purely imaginary, we must have $4a^3b - 4ab^3 = 0$, which simplifies to $ab(a^2 - b^2) = 0$. Therefore, either $a = 0$, $b = 0$, or $a^2 = b^2$.
If $a = 0$, then $z = bi$, and $|z|^2 = b^2$. Since $z^4$ is purely imaginary, we must have $Re(z^4) = 0$, which means $b^4$ = 0, so $b = 0$. Therefore, $z = 0$.
If $b = 0$, then $z = a$, and $|z|^2 = a^2$. Since $z^4$ is purely imaginary, we must have $Re(z^4) = 0$, which means $a^4$ = 0, so $a = 0$. Therefore, $z = 0$.
If $a^2 = b^2$, then we can write $z = a+ai$ or $z = a-ai$, depending on whether $a$ is positive or negative. In either case, $|z|^2 = 2a^2$, and we have:
\begin{align*}
z^4 &= (a+ai)^4 \\
&= (a^4 - 6a^2b^2 + b^4) + (4a^3b - 4ab^3)i \\
&= (a^4 - 3a^2b^2 + a^2b^2 + 2b^4) + (4a^3b - 4ab^3)i \\
&= (a^2 - b^2)^2 + 4a^2b^2i
\end{align*}
Since $z^4$ is purely imaginary, we must have $4a^2b^2 = 0$, which means $a = 0$ or $b = 0$. But $a^2 = b^2$, so we must have $a = b = 0$, which means $z = 0$.
Therefore, the only complex number in $S$ is $z = 0$.
Let $z = a+bi$ be a complex number in $S$. Then we have:
\begin{align*}
z^4 &= (a+bi)^4 \\
&= a^4 + 4a^3bi - 6a^2b^2 - 4ab^3i + b^4i^4 \\
&= (a^4 - 6a^2b^2 + b^4) + (4a^3b - 4ab^3)i
\end{align*}
Since $z^4$ is purely imaginary, we must have $4a^3b - 4ab^3 = 0$, which simplifies to $ab(a^2 - b^2) = 0$. Therefore, either $a = 0$, $b = 0$, or $a^2 = b^2$.
If $a = 0$, then $z = bi$, and $|z|^2 = b^2$. Since $z^4$ is purely imaginary, we must have $Re(z^4) = 0$, which means $b^4$ = 0, so $b = 0$. Therefore, $z = 0$.
If $b = 0$, then $z = a$, and $|z|^2 = a^2$. Since $z^4$ is purely imaginary, we must have $Re(z^4) = 0$, which means $a^4$ = 0, so $a = 0$. Therefore, $z = 0$.
If $a^2 = b^2$, then we can write $z = a+ai$ or $z = a-ai$, depending on whether $a$ is positive or negative. In either case, $|z|^2 = 2a^2$, and we have:
\begin{align*}
z^4 &= (a+ai)^4 \\
&= (a^4 - 6a^2b^2 + b^4) + (4a^3b - 4ab^3)i \\
&= (a^4 - 3a^2b^2 + a^2b^2 + 2b^4) + (4a^3b - 4ab^3)i \\
&= (a^2 - b^2)^2 + 4a^2b^2i
\end{align*}
Since $z^4$ is purely imaginary, we must have $4a^2b^2 = 0$, which means $a = 0$ or $b = 0$. But $a^2 = b^2$, so we must have $a = b = 0$, which means $z = 0$.
Therefore, the only complex number in $S$ is $z = 0$.
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For a complex number z, let Re(z) denote the real part of z. Let Sbe the set of all complex numbers zsatisfying z4−|z|4 = 4iz2, where i= √-1. Then the minimum possible value of |z1-z2|2, where z1, z2 ∈Swith Re(z1) > 0 and Re(z2) < 0, is _____Correct answer is '8'. Can you explain this answer?
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For a complex number z, let Re(z) denote the real part of z. Let Sbe the set of all complex numbers zsatisfying z4−|z|4 = 4iz2, where i= √-1. Then the minimum possible value of |z1-z2|2, where z1, z2 ∈Swith Re(z1) > 0 and Re(z2) < 0, is _____Correct answer is '8'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about For a complex number z, let Re(z) denote the real part of z. Let Sbe the set of all complex numbers zsatisfying z4−|z|4 = 4iz2, where i= √-1. Then the minimum possible value of |z1-z2|2, where z1, z2 ∈Swith Re(z1) > 0 and Re(z2) < 0, is _____Correct answer is '8'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For a complex number z, let Re(z) denote the real part of z. Let Sbe the set of all complex numbers zsatisfying z4−|z|4 = 4iz2, where i= √-1. Then the minimum possible value of |z1-z2|2, where z1, z2 ∈Swith Re(z1) > 0 and Re(z2) < 0, is _____Correct answer is '8'. Can you explain this answer?.
For a complex number z, let Re(z) denote the real part of z. Let Sbe the set of all complex numbers zsatisfying z4−|z|4 = 4iz2, where i= √-1. Then the minimum possible value of |z1-z2|2, where z1, z2 ∈Swith Re(z1) > 0 and Re(z2) < 0, is _____Correct answer is '8'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about For a complex number z, let Re(z) denote the real part of z. Let Sbe the set of all complex numbers zsatisfying z4−|z|4 = 4iz2, where i= √-1. Then the minimum possible value of |z1-z2|2, where z1, z2 ∈Swith Re(z1) > 0 and Re(z2) < 0, is _____Correct answer is '8'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For a complex number z, let Re(z) denote the real part of z. Let Sbe the set of all complex numbers zsatisfying z4−|z|4 = 4iz2, where i= √-1. Then the minimum possible value of |z1-z2|2, where z1, z2 ∈Swith Re(z1) > 0 and Re(z2) < 0, is _____Correct answer is '8'. Can you explain this answer?.
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