The locus of the variable point $\mathrm{z}=\mathrm{x}+\mathrm{iy}$ whose amplitude is always equal to…

The locus of the variable point $\mathrm{z}=\mathrm{x}+\mathrm{iy}$ whose amplitude is always equal to $\theta$, is
  1. $\mathrm{x}^2+\mathrm{y}^2=\tan ^2 \theta$
  2. $y=x \tan \theta$
  3. $\frac{x^2}{\sin ^2 \theta}+\frac{y^2}{\cos ^2 \theta}=1$
  4. $\frac{x^2}{\sin ^2 \theta}-\frac{y^2}{\cos ^2 \theta}=1$

Solution

Given $\arg (z)=\theta=\tan ^{-1}\left(\frac{y}{x}\right) \Rightarrow \frac{y}{x}=\tan \theta$ $\Rightarrow y=x \tan \theta$

Asked in: AP EAMCET 2023 (17 May Shift 2)

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