Q.

Of all complex numbers z satisfying z-4z=3, let α and β be the complex numbers having the greatest and the least possible modulus respectively.  Denote by Re(z) and Im(z) the real and imaginary parts respectively of the  complex number z . Which of the following statement(s) is/are false?

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a

|α| equals 4+22

b

|β| equals 4-22

c

Imα=0

d

Reβ=0

answer is A.

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Detailed Solution

since z1-z2≤z1-z2 ⇒z-4z≤z-4z=3   ⇒r-4r≤3 , where r=z ⇒-3≤r-4r≤3 ⇒ r2+3r-4≥0 and r2-3r-4≤0 ⇒r≥1 and -1≤r≤4 ⇒there1≤ r ≤4    ⇒ r =z maximum value =4  and z minimum value =1      and z=4, z=1 satisfies the given equation whose imaginary part is 0 hence options 1,2,4 are wrong
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