Question

Question 12 Let z1= 2-i , z2= -2+i. Find

(i) Re(z1z2/z1)

(ii) Im( 1/ z1.z1)

Class X1 - Maths -Complex Numbers and Quadratic Equations Page 113

Answer 1

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