Question

If x,y,b are real and z-i/z-1 = ib, show that (x-1/2)² +(y-1/2)²= 1/2.​

Answer 1

Answer:

It is given that,

z-i/z-1=ib

where x,y,b are real..

As we know,

z=x+iy

Substitute the z value in the equation,

x+iy-i/x+iy-1=ib

x+i(y-1)/(x-1)+iy=ib

on rationalizing,

x+i(y-1)/(x-1)+iy×(x-1)-iy/(x-1)-iy=ib

x²-x-xiy+i(xy-y-x+1)+y²-y/(x-1)²+y²=ib

x²-x-y+y²+i(-x-y+1)/(x-1) ²+y²=ib

Comparing real and imaginary parts,

x²-x-y+y²/(x-1)²+y²=0

x²+y²-x-y=0

x²+y²=x+y

In the above question,

Take LHS,

(x-1/2)²+(y-1/2)²

x²+1/4-x+y²+1/4-y

since, x²+y²=x+y

x+y+1/4-x-y+1/4

=1/4+1/4

=2/4=1/2

so,LHS=RHS

Step-by-step explanation:

Hope it helps you frnd...