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Complex Numbers question

2020 · 5 Sep · Shift 1 · Q34
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  5. /2020 · 5 Sep · Shift 1 · Q34

Complex Numbers question

2020 · 5 Sep · Shift 1 · Q34

JEE MainMathematicsComplex NumbersMCQ+4 / −1
If the four complex numbers z,z‾,z‾−2Reolimits(z‾)z,\overline z ,\overline z - 2{\mathop{\rm Re} olimits} \left( {\overline z } \right)z,z,z−2Reolimits(z) and z−2Re(z)z-2Re(z)z−2Re(z) represent the vertices of a square of side 4 units in the Argand plane, then ∣z∣|z|∣z∣ is equal to :
  1. A
    4 2\sqrt 22​
  2. B
    4
  3. C
    2
  4. D
    2 2\sqrt 22​
View written solutionFree

Correct answer: D

  1. Let z=x+iyz=x+iyz=x+iy where x,y∈Rx,y\in \mathbb{R}x,y∈R.

  2. Then the given four complex numbers become:

    • z=x+iyz=x+iyz=x+iy
    • z‾=x−iy\overline z=x-iyz=x−iy
    • z‾−2Re⁡(z‾)=(x−iy)−2x=−x−iy\overline z-2\operatorname{Re}(\overline z)=(x-iy)-2x=-x-iyz−2Re(z)=(x−iy)−2x=−x−iy
    • z−2Re⁡(z)=(x+iy)−2x=−x+iyz-2\operatorname{Re}(z)=(x+iy)-2x=-x+iyz−2Re(z)=(x+iy)−2x=−x+iy

  3. So the four points in the Argand plane are: (x,y), (x,−y), (−x,−y), (−x,y)(x,y),\ (x,-y),\ (-x,-y),\ (-x,y)(x,y), (x,−y), (−x,−y), (−x,y)

  4. These are the vertices of a rectangle centered at the origin, with side lengths:

    • horizontal side =2∣x∣=2|x|=2∣x∣
    • vertical side =2∣y∣=2|y|=2∣y∣

    Since they form a square, we must have 2∣x∣=2∣y∣  ⟹  ∣x∣=∣y∣.2|x|=2|y|\implies |x|=|y|.2∣x∣=2∣y∣⟹∣x∣=∣y∣.

  5. The side of the square is given as 444. Hence 2∣x∣=4and2∣y∣=42|x|=4 \quad \text{and} \quad 2|y|=42∣x∣=4and2∣y∣=4 so ∣x∣=∣y∣=2.|x|=|y|=2.∣x∣=∣y∣=2.

  6. Therefore, ∣z∣=x2+y2=22+22=8=22.|z|=\sqrt{x^2+y^2}=\sqrt{2^2+2^2}=\sqrt{8}=2\sqrt{2}.∣z∣=x2+y2​=22+22​=8​=22​.

  7. Option check:

    • A: 424\sqrt242​ ❌
    • B: 444 ❌
    • C: 222 ❌
    • D: 222\sqrt222​ ✅

Hence the correct answer is D.

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