JEE Main 2022 · 26 Jun · Shift 1 · Mathematics Q2

← JEE Main 2022 · 26 Jun · Shift 1

MathematicsSingle correct+4 / -1Question 2 of 90

Let $A=\left\{z\in\mathbb C:\left|\dfrac{z+1}{z-1}\right|<1\right\}$ and $B=\left\{z\in\mathbb C:\arg\left(\dfrac{z-1}{z+1}\right)=\dfrac{2\pi}3\right\}$ .

Then $A\cap B$ is :

a portion of a circle centred at

\left(0,-\dfrac1{\sqrt3}\right)

that lies in the second and third quadrants only

a portion of a circle centred at

\left(0,-\dfrac1{\sqrt3}\right)

that lies in the second quadrant only

an empty set

a portion of a circle of radius

\dfrac2{\sqrt3}

that lies in the third quadrant only

Answer and worked solution

Official answer: B

Set A. $|z+1|<|z-1|$ means $z$ is nearer to $-1$ than to $1$ : $\mathrm{Re}\,z<0$ .

Set B. The segment from $-1$ to $1$ subtends $\frac{2\pi}3$ at $z$ , so $z$ lies on an arc of a circle through $\pm1$ . The positive argument puts the arc above the real axis, and an obtuse inscribed angle puts the centre below: centre $\left(0,-\frac1{\sqrt3}\right)$ , radius $\frac2{\sqrt3}$ .

Intersection. The part of that upper arc with $\mathrm{Re}\,z<0$ : in the second quadrant only.

(B)

Topics:Geometry in the complex plane Modulus and argument

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JEE Main 2022 · 26 Jun · Shift 1, Mathematics question 2