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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Complex Number

NDA 2021 · Mathematics · Complex Number

If z=x+i y , where i= -1 , then what does the equation z z...

If z=x+iyz=x+i y, where i=−1i=\sqrt{-1}, then what does the equation zzˉ+∣z∣2+4(z+zˉ)−48=0z \bar{z}+|z|^2+4(z+\bar{z})-48=0 represent?

Options

  1. AStraight line
  2. BParabola
  3. CCircle
  4. DPair of straight lines

Correct answer

C. Circle

Step-by-step solution

Here z=x+iy⇒zˉ=x−iyz=x+i y \Rightarrow \bar{z}=x-i y∴zzˉ+∣z∣2+4(z+zˉ)−48=0⇒∣z∣2+∣z∣2+4(z+zˉ)−48=0[∴zzˉ=∣z∣2]⇒2(x2+y2)+4(2x)−48=0⇒x2+y2+4x−48=0\begin{aligned} \therefore & z \bar{z}+|z|^2+4(z+\bar{z})-48=0 \\ & \Rightarrow|z|^2+|z|^2+4(z+\bar{z})-48=0\left[\therefore z \bar{z}=|z|^2\right] \\ & \Rightarrow 2\left(x^2+y^2\right)+4(2 x)-48=0 \\ & \Rightarrow x^2+y^2+4 x-48=0 \end{aligned}$$ Represent equation of circles.
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