JEE Main · 2022 · 29 Jun 2022 · Shift 2 · Complex Numbers

Let arg

(z)

represent the principal argument of the complex number

z

. Then,

∣ z ∣ = 3

and

ar g (z - 1) - ar g (z + 1) = \frac{π}{4}

intersect:

Answer: (C)

Solution

The condition

∣ z ∣ = 3

represents a circle of radius 3 centered at the origin. The condition

ar g (z - 1) - ar g (z + 1) = \frac{π}{4}

represents the locus of points from which the segment joining

- 1

and

1

subtends an angle of

\frac{π}{4}

. This is an arc of a circle passing through

- 1

and

1

. By the inscribed angle theorem, this arc lies on a circle with a specific radius. Computing the radius of this circle and comparing with the circle

∣ z ∣ = 3

, we find that these two circles do not intersect. Therefore, the two loci have no common points.

JEE Main · 2022 · 29 Jun 2022 · Shift 2 · Single correct option