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Let z be a complex number such that $|z+2|=|z-2|$ and $\arg \left(\frac{z+3}{z-i}\right)=\frac{\pi}{4}$. Then $|z|^{2}$ is equal to

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  • 22 October, 2026 at 6:00 pm #12273
    UPLIFTJEE
    Keymaster

    Let z be a complex number such that $|z+2|=|z-2|$ and $\arg \left(\frac{z+3}{z-i}\right)=\frac{\pi}{4}$. Then $|z|^{2}$ is equal to
    [JEE MAIN 04 April 2026 S I]
    (A) 9
    (B) 4
    (C) 5
    (D) 1

    22 October, 2026 at 6:00 pm #12274
    UPLIFTJEE
    Keymaster

    Solution

    $|\mathrm{z}+2|=|\mathrm{z}-2|$ let $\mathrm{z}=\mathrm{x}+\mathrm{iy}$

    \[ \begin{aligned} & |x+2+i y|=|x-2+i y| \Rightarrow(x+2)^{2}+y^{2}=(x-2)^{2}+y^{2} \Rightarrow x=0 \\ & \Rightarrow z=0+i y ; y \in R \\ & \arg \left(\frac{3+i y}{i y-i}\right)=\arg \left(\frac{y}{y-1}+\frac{3 i}{1-y}\right)=\frac{\pi}{4} \\ & \Rightarrow \frac{y}{y-1}=\frac{3}{1-y}>0 \Rightarrow y=-3 \Rightarrow z=-3 i \Rightarrow|z|^{2}=9 \end{aligned} \]
    22 October, 2026 at 6:00 pm #12275
    UPLIFTJEE
    Keymaster

    Answer

    A

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