Question

The equation $r=\left|\cos \theta \right|$ represents

• A. two circle of radii $\frac{1}{2}$ each
• B. two circles centered at $(\frac{1}{2},0)$ and $(-\frac{1}{2},0)$
• C. two circles touching each other at the origin
• D. pair of straight line

Answer 1

Answer: (A), (B), (C)

The correct options are
A two circle of radii $\frac{1}{2}$ each
B two circles centered at $(\frac{1}{2},0)$ and $(-\frac{1}{2},0)$
C two circles touching each other at the origin

Multiplying by $r,$ the given equation become $r^2=\left|r\cos \theta \right|$ ...(1)

In Cartesian form equation (1) can be written as $x^2+y^2=\left|x\right|$

$\Rightarrow x^2+y^2=+x$ if $x>0$

and $x^2+y^2=-x$ if $x<0$

Equation (2) represents a circle of radius $\frac{1}{2}$ and centered at $(\frac{1}{2},0).$

Equation (3) represents a circle of radius $\frac{1}{2}$ and centered at $(-\frac{1}{2},0).$

Hence, the given equation represents two circles touching each other at the origin.