JEE Main 2024MathematicsComplex NumberHardNumerical

JEE Main 2024 — Complex Number Question with Solution

JEE Main 2024 (27 Jan Shift 2)

Question

Let the complex numbers α and 1α¯ lie on the circles z-z02=4 and z-z02=16 respectively, where z0=1+i. Then, the value of 100|α|2 is__________.

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Show full solutionCorrect answer: 20
Correct answer
20

Step-by-step explanation

Given: z-z02=4

⇒z-z0z-z0=4

⇒zz-z0z-z0z+z02=4

⇒z2-z0z-z0z=2 as z02=2

It is given that, α lies on z-z02=4.

⇒α2-αz0-z0α-2=0   ...i

We know that, α·α=α2=α2

z-z02=16

⇒1α¯-z01α-z¯0=16

⇒1-α¯z01-αz¯0=16|α|2

⇒1-α¯z0-αz¯0+|α|2z02=16|α|2

⇒1-α¯z0-αz¯0=14|α|2   ...ii

Using, i and ii

⇒5|α|2=1

⇒100|α|2=20

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About this question

This is a previous-year question from JEE Main 2024, covering the Complex Number chapter of Mathematics. PrepSharp catalogues every PYQ from JEE Main with a verified answer key and step-by-step solution prepared by IIT alumni — so you can search by chapter, topic or year and revise efficiently.