Question

For a complex number z, If Re(z) and Im(z) are the roots of x^2 7x 120
and Z+Z is one of the roots of x^2 - 10x + 160, then find Re(2) -Im(2).

Answer 1

Let z=x+iy,x,y∈R and i=

−1

∵Re(z)=∣z−2∣

⇒x=

(x−2)

2

+y

2

⇒x

2

=(x−2)

2

+y

2

⇒x

2

=x

2

−4x+4+y

2

⇒y

2

=4(x−1)

∴z=1+t

2

+2ti, parametric form and let ω=p+iq

Similarly,ω=1+s

2

+2si

∴z−ω=(t

2

−s

2

)+2i(t−s)

⇒arg(z−ω)=

3

π

∴tan

−1

(

t+s

2

)=

3

π

⇒

t+s

2

=

3

Now, z+ω=2+t

2

+s

2

+2i(t+s)

∴Im(z+ω)=2(t+s)=

3

4