JEE Advanced · Mathematics · 3. Complex Numbers

Let $\bar{z}$ denote the complex conjugate of a complex number $z$ . If $z$ is a non-zero complex number for which both real and imaginary parts of ${(\bar{z})}^{2} + \frac{1}{z^{2}}$ are integers, then which of the following is/are possible value(s) of $|z|$ ?

A ${(\frac{43 + 3 \sqrt{205}}{2})}^{\frac{1}{4}}$
B ${(\frac{7 + \sqrt{33}}{4})}^{\frac{1}{4}}$
C ${(\frac{9 + \sqrt{65}}{4})}^{\frac{1}{4}}$
D ${(\frac{7 + \sqrt{13}}{6})}^{\frac{1}{4}}$

Verified Solution

Answer & Solution

Correct Answer

(A) ${(\frac{43 + 3 \sqrt{205}}{2})}^{\frac{1}{4}}$

Step-by-step Solution

Detailed explanation

Let

z = r . e^{i θ}

So,

{(\bar{z})}^{2} + \frac{1}{z^{2}} = (r^{2} + \frac{1}{r^{2}}) e^{- 2 i θ} = a + i b

(say), given

a, b \in Z

Now taking mod both side and squaring we get,

{(r^{2} + \frac{1}{r^{2}})}^{2} = a^{2} + b^{2}

\Rightarrow r^{8} - (a^{2} + b^{2} - 2) r^{4} + 1 = 0

\Rightarrow r^{4} = \frac{(a^{2} + b^{2} - 2) \pm \sqrt{{(a^{2} + b^{2} - 2)}^{2} - 4}}{2}

Now comparing from option we get, for

a^{2} + b^{2} = 45

(i.e

(a, b) = (\pm 6, \pm 3)

(\pm 3, \pm 6)

We get

r = {(\frac{43 + 3 \sqrt{205}}{2})}^{\frac{1}{4}}

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.

Same subject

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

Column 1	Column 2	Column 3
(I) $x^{2} + y^{2} = a^{2}$	(i) $m y = m^{2} x + a$	(P) $(\frac{a}{m^{2}}, \frac{2 a}{m})$
(II) $x^{2} + a^{2} y^{2} = a^{2}$	(ii) $y = m x + a \sqrt{m^{2} + 1}$	(Q) $(\frac{- m a}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}})$
(III) $y^{2} = 4 a x$	(iii) $y = m x + \sqrt{a^{2} m^{2} - 1}$	(R) $(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{1}{\sqrt{a^{2} m^{2} + 1}})$
(IV) $x^{2} - a^{2} y^{2} = a^{2}$	(iv) $y = m x + \sqrt{a^{2} m^{2} + 1}$	(S) $(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} - 1}}, \frac{- 1}{\sqrt{a^{2} m^{2} - 1}})$

For

a = \sqrt{2}

, if a tangent is drawn to a suitable conic (Column 1) at the point of contact

(- 1, 1)

, then which of the following options is the only Correct combination for obtaining its equation?

JEE Advanced 2017 Hard

Consider the set of eight vectors $V = \{a \hat{i} + b \hat{j} + c \hat{k} : a, b, c \in \{- 1, 1\}\}$ . Three non-coplanar vectors can be chosen from $V$ in $2^{p}$ ways. Then $p$ is

JEE Advanced 2013 Medium

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Let z¯ denote the complex conjugate of a complex number z. If z is a non-zero complex number for which both real and imaginary parts of z¯2+1z2 are integers, then which of the following is/are possible value(s) of z ?

Step-by-step Solution

Let $\bar{z}$ denote the complex conjugate of a complex number $z$ . If $z$ is a non-zero complex number for which both real and imaginary parts of ${(\bar{z})}^{2} + \frac{1}{z^{2}}$ are integers, then which of the following is/are possible value(s) of $|z|$ ?