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Let z be a complex number such that $\left|\frac{z-2 i}{z+i}\right|=2, z \neq-i$. Then z lies on the circle of radius 2 and centre

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  • 30 October, 2026 at 8:11 am #12544
    UPLIFTJEE
    Keymaster

    Let z be a complex number such that $\left|\frac{z-2 i}{z+i}\right|=2, z \neq-i$. Then z lies on the circle of radius 2 and centre
    [JEE (Main) 25 Jan 2023 S II], [JEE (Main) 25 Jan 2023 S II]
    (A) $(0,0)$
    (B) (0, -2)
    (C) $(2,0)$
    (D) $(0,2)$

    30 October, 2026 at 8:11 am #12545
    UPLIFTJEE
    Keymaster

    Solution

    $(\mathrm{z}-2 \mathrm{i})(\overline{\mathrm{z}}+2 \mathrm{i})=4(\mathrm{z}+\mathrm{i})(\overline{\mathrm{z}}-\mathrm{i})$

    \[ \begin{aligned} & z \bar{z}+4+2 i(z-\bar{z})=4(z \bar{z}+1+i(\bar{z}-z)) \\ & 3 z \bar{z}-6 i(z-\bar{z})=0 \Rightarrow x^{2}+y^{2}-2 i(2 i y)=0 \Rightarrow x^{2}+y^{2}+4 y=0 \end{aligned} \]
    30 October, 2026 at 8:11 am #12546
    UPLIFTJEE
    Keymaster

    Answer

    B

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