Question

If `|(z - 2)/(z + 2)| = pi/6`, then the locus of z is ______.

Answer 1

If `|(z - 2)/(z + 2)| = pi/6`, then the locus of z is circle.

Explanation:

Given that: `|(z - 2)/(z + 2)| = pi/6`

Let z = x + iy

⇒ `|(x + iy - 2)/(x + iy + 2)| = pi/6`

⇒ `|((x - 2) + iy)/((x + 2) + iy)| = pi/6`

⇒ `6|(x - 2) + iy| = pi|(x + 2) + iy|`

⇒ `6sqrt((x - 2)^2 + y^2) = pisqrt((x + 2)^2 + y^2)`

⇒ `36[x^2 + 4 - 4x + y^2] = pi^2[x^2 + 4 + 4x + y^2]`

⇒ 36x² + 144 – 144x + 36y² = π²x² + 4π² + 4π²x + π²y2

⇒ (36 – π²)x² + (36 – π²)y2 – (144 + 4π²)x + 144 – 4π² = 0

Which represents are equation of a circle.