Question

JEE ADVANCED MATHS QUESTION SEPTEMBER 2020

Answer 1

For a complex number z, let Re(z) denotes the real part of z. let S be the set of all complex z satisfying z⁴ - |z|⁴ = 4 iz², where I = √-1. then the minimum possible value of |z₁ - z₂|² where z₁ and z₂ ∈ S with Re(z₁) > 0 and Re(z₂) < 0, is...

solution : complex equation is z⁴ - |z|⁴ = 4i z²

⇒z⁴ - (|z|²)² = 4i z²

⇒z⁴ - (z bar(z))² = 4i z²

⇒z⁴ - z²bar(z)² = 4i z²

⇒z²(z² - bar(z)²) = 4i z²

⇒(z + bar(z))(z - bar(z)) = 4i

⇒[(z + bar(z))/2] [(z - bar(z))/2i] = 1

let (z + bar(z))/2 = x and (z - bar(z))/2i = y

then, xy = 1 [ a hyperbolic equation ]

see the diagram, here it is clear that minimum value of |z1 - z2| possible only when x, y ∈ {(1, 1) , (-1,-1)}

so, z = 1 + i and z2 = -1 - i

then, |z1 - z2|² = |1 + i + 1 + i|²

= |2 + 2i|² = (2)² + (2)² = 4 + 4 + 8

Therefore the value of |z1 - z2| is 8