Let and . Suppose , where . If and , then lies…

MATHEMATICS · Single correct · CBSE · Mathematics · Complex Numbers and Quadratic Equations · Complex Numbers and Quadratic Equations

Problem

Let

a, b \in \mathbb{R}

and

a^2 + b^2 \neq 0

. Suppose

S = \left\{ z \in \mathbb{C} : z = \frac{1}{a+ibt}, t \in \mathbb{R}, t \neq 0 \right\}

, where

i = \sqrt{-1}

. If

z = x + iy

and

z \in S

, then

(x, y)

lies on (1) the circle with radius

\frac{1}{2a}

and centre

\left(\frac{1}{2a}, 0\right)

for

a > 0, b \neq 0

(2) the circle with radius

-\frac{1}{2a}

and centre

\left(-\frac{1}{2a}, 0\right)

for

a < 0, b \neq 0

(3) the

X

-axis for

a \neq 0, b = 0

(4) the

Y

-axis for

a = 0, b \neq 0