Which of the following equations gives a circle?

Which of the following equations gives a circle?
  1. $r=2 \sin \theta$
  2. $r^2 \cos 2 \theta=1$
  3. $r(4 \cos \theta+5 \sin \theta)=3$
  4. $5=r(1+\sqrt{2} \cos \theta)$

Solution

Now, taking option (a) i.e. $r=2 \sin \theta$ Let $x=r \sin \theta, y=r \cos \theta$ $\Rightarrow \quad r^2=2 r \sin \theta$ $\Rightarrow \quad x^2+y^2=2 y \quad\left[\because x^2+y^2=r^2\right]$ This equation represents a circle.

Asked in: AP EAMCET 2005

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