If `z=x+i y ,` then show that `z bar z +2(z+ bar z )+a=0` , where `a in R ,` represents a circle.

Explanation

To show that the equation \( z \bar{z} + 2(z + \bar{z}) + a = 0 \) represents a circle, we start by expressing \( z \) in terms of its real and imaginary parts. Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

### Step 1: Express \( z \bar{z} \)
The conjugate of \( z \) is given by:
\[ \bar{z} = x - iy \]
Now, we can compute \( z \bar{z} \):
...