If z = x + iy, then show that zz̅ + 2(z + z̅) + b = 0 where bϵR, re

1.	If z = x + iy, then show that zz̅ + 2(z + z̅) + b = 0 where bϵR, representing z in the complex plane is a circle.
Answer» According to the question, We have, z = x + iy ⇒ z̅ = x – iy Now, we also have, z z̅ + 2 (z + z̅) + b = 0 ⇒ (x + iy) (x – iy) + 2 (x + iy + x – iy) + b = 0 ⇒ x² + y² + 4x + b = 0 The equation obtained represents the equation of a circle.

If z = x + iy, then show that zz̅ + 2(z + z̅) + b = 0 where bϵR, representing z in the complex plane is a circle.

Answer»

According to the question,

We have,

z = x + iy

⇒ z̅ = x – iy

Now, we also have,

z z̅ + 2 (z + z̅) + b = 0

⇒ (x + iy) (x – iy) + 2 (x + iy + x – iy) + b = 0

⇒ x² + y² + 4x + b = 0

The equation obtained represents the equation of a circle.