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Complex Numbers question

2022 · 26 Jul · Shift 2 · Q21
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Complex Numbers question

2022 · 26 Jul · Shift 2 · Q21

JEE MainMathematicsComplex NumbersMCQ+4 / −1
If z=x+iyz=x+i yz=x+iy satisfies ∣z∣−2=0|z|-2=0∣z∣−2=0 and ∣z−i∣−∣z+5i∣=0|z-i|-|z+5 i|=0∣z−i∣−∣z+5i∣=0, then :
  1. A
    x+2y−4=0x+2 y-4=0x+2y−4=0
  2. B
    x2+y−4=0x^{2}+y-4=0x2+y−4=0
  3. C
    x+2y+4=0x+2 y+4=0x+2y+4=0
  4. D
    x2−y+3=0x^{2}-y+3=0x2−y+3=0
View written solutionFree

Correct answer: C

  1. Let z=x+iyz=x+iyz=x+iy with real x,yx,yx,y.

We are given: ∣z∣−2=0 ⇒ ∣z∣=2|z|-2=0 \,\Rightarrow\, |z|=2∣z∣−2=0⇒∣z∣=2 and ∣z−i∣−∣z+5i∣=0⇒∣z−i∣=∣z+5i∣.|z-i|-|z+5i|=0 \Rightarrow |z-i|=|z+5i|.∣z−i∣−∣z+5i∣=0⇒∣z−i∣=∣z+5i∣.

  1. Use the first condition.

Since ∣z∣=x2+y2=2,|z|=\sqrt{x^2+y^2}=2,∣z∣=x2+y2​=2, we get x2+y2=4.(1)x^2+y^2=4. \qquad (1)x2+y2=4.(1)

  1. Use the second condition.

Now, z−i=x+i(y−1)z-i=x+i(y-1)z−i=x+i(y−1) so ∣z−i∣=x2+(y−1)2.|z-i|=\sqrt{x^2+(y-1)^2}.∣z−i∣=x2+(y−1)2​.

Also, z+5i=x+i(y+5)z+5i=x+i(y+5)z+5i=x+i(y+5) so ∣z+5i∣=x2+(y+5)2.|z+5i|=\sqrt{x^2+(y+5)^2}.∣z+5i∣=x2+(y+5)2​.

Given these are equal: x2+(y−1)2=x2+(y+5)2.\sqrt{x^2+(y-1)^2}=\sqrt{x^2+(y+5)^2}.x2+(y−1)2​=x2+(y+5)2​.

Squaring both sides, x2+(y−1)2=x2+(y+5)2.x^2+(y-1)^2=x^2+(y+5)^2.x2+(y−1)2=x2+(y+5)2.

So, (y−1)2=(y+5)2.(y-1)^2=(y+5)^2.(y−1)2=(y+5)2.

Expand: y2−2y+1=y2+10y+25.y^2-2y+1=y^2+10y+25.y2−2y+1=y2+10y+25.

Thus, −2y+1=10y+25-2y+1=10y+25−2y+1=10y+25 −12y=24-12y=24−12y=24 y=−2.y=-2.y=−2.

  1. Substitute into equation (1): x2+y2=4x^2+y^2=4x2+y2=4 x2+(−2)2=4x^2+(-2)^2=4x2+(−2)2=4 x2+4=4x^2+4=4x2+4=4 x2=0⇒x=0.x^2=0 \Rightarrow x=0.x2=0⇒x=0.

Hence, z=−2i.z= -2i.z=−2i.

  1. Check the options.

For x=0,y=−2x=0, y=-2x=0,y=−2:

  • A: x+2y−4=0+2(−2)−4=−8≠0x+2y-4=0+2(-2)-4=-8 \neq 0x+2y−4=0+2(−2)−4=−8=0 False.

  • B: x2+y−4=0−2−4=−6≠0x^2+y-4=0-2-4=-6 \neq 0x2+y−4=0−2−4=−6=0 False.

  • C: x+2y+4=0+2(−2)+4=0x+2y+4=0+2(-2)+4=0x+2y+4=0+2(−2)+4=0 True.

  • D: x2−y+3=0−(−2)+3=5≠0x^2-y+3=0-(-2)+3=5 \neq 0x2−y+3=0−(−2)+3=5=0 False.

Therefore, the correct option is C.\boxed{\text{C}}.C​.

  1. Comparison with stored answer:

Stored correct answer: C

This matches the derived answer.

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