Let a, b ∈R and a2+b2≠0 . Suppose S=z∈C :z=1a+ibt, t ∈R, t ≠0, where i= -1. If z=x+iy and z∈S, then x, y lies on

(A) The circle with radius 12a and centre 12a, 0 for a>0, b ≠0

(B) The circle with radius -12a and centre -12a, 0 for a<0, b≠0

(C) The x - axis for a ≠0, b=0

(D) The y - axis for a=0, b ≠0

JEE Advanced · Mathematics · 3. Complex Numbers

Let $a, b \in R$ and $a^{2} + b^{2} \neq 0$ . Suppose $S = \{z \in C : z = \frac{1}{a + i b t}, t \in R, t \neq 0\},$ where $i = \sqrt{- 1} .$ If $z = x + i y$ and $z \in S,$ then $(x, y)$ lies on

A The circle with radius $\frac{1}{2 a}$ and centre $(\frac{1}{2 a}, 0)$ for $a > 0, b \neq 0$
B The circle with radius $- \frac{1}{2 a}$ and centre $(- \frac{1}{2 a}, 0)$ for $a < 0, b \neq 0$
C The x - axis for $a \neq 0, b = 0$
D The y - axis for $a = 0, b \neq 0$

Verified Solution

Answer & Solution

Correct Answer

(A) The circle with radius $\frac{1}{2 a}$ and centre $(\frac{1}{2 a}, 0)$ for $a > 0, b \neq 0$

Step-by-step Solution

Detailed explanation

x + i y = \frac{1}{a + i b t}

(Multiply & divide by conjugate

a + i b t

)

x + i y = \frac{a - i b t}{a^{2} + b^{2} t^{2}}

Let

a \neq 0

and

b \neq 0

x = \frac{a}{a^{2} + b^{2} t^{2}}

......(i)

y = \frac{- b t}{a^{2} + b^{2} t^{2}}

......(ii)

\frac{y}{x} = \frac{- b t}{a} \Rightarrow t = - \frac{a y}{b x}

Put in (i)

x \{a^{2} + b^{2} . \frac{a^{2} y^{2}}{b^{2} x^{2}}\} = a

\Rightarrow

a^{2} (x^{2} + y^{2}) = a x

x^{2} + y^{2} - \frac{1}{a} x = 0

{(x - \frac{1}{2 a})}^{2} + y^{2} = \frac{1}{4 a^{2}}

Circle with centre

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

, radius

= \frac{1}{2 | a |}

{\begin{matrix} a > 0; \frac{1}{2 a} \\ a < 0; \frac{- 1}{2 a} \end{matrix}

For

a \neq 0, b = 0

x + i y = \frac{1}{a}

x = \frac{1}{a}, y = 0 \Rightarrow z

lies on x - axis
For

a = 0, b \neq 0

x + i y = \frac{1}{i b t}

y = - \frac{1}{b t}, x = 0

\Rightarrow

z lies on y - axis.

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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