The closest distance of the origin from a curve given as $$a\bar z + \bar az + a\bar a = 0$$ ($$a$$ is a complex number) is.
  • A
    $$1$$
  • A
    $$1$$
  • B
    $$\dfrac {Re(|a|)}{|a|}$$
  • B
    $$\dfrac {Re(|a|)}{|a|}$$
  • C
    $$\dfrac {Im(|a|)}{|a|}$$
  • C
    $$\dfrac {Im(|a|)}{|a|}$$
  • D
    $$\dfrac {|a|}{2}$$
  • D
    $$\dfrac {|a|}{2}$$

Solution

The closest distance $$=$$ length of perpendicular from the origin on the line $$a\bar z + \bar az + a\bar a = 0$$ 

$$\begin{array}{l} =\frac { { a\left( 0 \right) +\bar { a } \left| 0 \right| +a\bar { a }  } }{ { 2\left| a \right|  } }  \\ =\frac { { { { \left| a \right|  }^{ 2 } } } }{ { 2\left| a \right|  } }  \\ =\frac { { \left| a \right|  } }{ 2 }  \end{array}$$

Hence, Option $$D$$ is the correct answer.