Let z 1 and z 2 be two complex numbers such that arg z 1 - z 2 = π 4 and z 1 , z 2 satisfy the equation…

Let z1 and z2 be two complex numbers such that argz1-z2=π4 and z1,z2 satisfy the equation z-3=Re(z). Then the imaginary part z1+z2 is equal to

Solution

Let z1=x1+iy1,z2=x2+iy2
z1-z2=x1-x2+iy1-y2
∵argz1-z2=π4⇒tan-1y1-y2x1-x2=π4
⇒y1-y2=x1-x2----(1)
z1-3=Rez1⇒x1-32+y12=x12
z2-3=Rez2⇒x2-32+y22=x22
⇒x1-32-x2-32+y12-y22=x12-x22
⇒x1-x2x1+x2-6+y1-y2y1+y2=x1+x2x1-x2
⇒x1+x2-6+y1+y2=x1+x2
⇒y1+y2=6

Asked in: JEE Main 2021 (27 Aug Shift 2)

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