Question

ifZ1= 2+i, Z2 = - 2+i, , then the imaginary part of​

Answer 1

Here's Your Answer Mate!

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z1 = 2 - i

z2 = -2 + i

(1) Re(z1.z2/conjugate of z1)

first of all we will solve and convert z1.z2/(conj of z1) in the form of (a + ib) by putting the given values . after then take real part (Re)

Z1.z2/conj of z1 = (2-i)(-2+i)/(2+i)

= -(2 - i)(2 - i)/(2 + i)

= -(2 - i)²/(2 + i)

= -(4 + i² -4i)/(2 + i)

= -(4 -1 -4i)/(2 + i)

= (-3 + 4i)/(2 + i)

Now, multiply with (2-i) both sides,

= (-3 + 4i)(2 - i)/(2 +i)(2-i)

= (-6 +3i+8i-4i²)/(2²-i²)

= (-2 + 11i)/(4 +1)

= (-2 + 11i)/5

= (-2/5) + (11/5)i

Now,

Re(z1.z2/conj of z1) = -2/5

(ii) Im(1/z1.conj of z1)

we know,

z1.conj of z1 = |z1|² use this concept here,

Im(1/z1.conj of z2) = Im(1/|z1|²)

Now,

|z1|² = |-2 + i|² = (√(4 +1)² = 5 + 0.i

Hence , Im(1/z1.conj of z1) = 0

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