1.

Solve the system of equations `R e(z^2)=0, |z|=2`

Answer» Given that, Re `(z^(2)) = 0, |z| = 2`
Let `z = x + iy`
`|z|=sqrt(x^(2) + y^(2))`
`:. sqrt(x^(2) + y^(2)) = 2`
`rArr x^(2) + y^(2) = 4 " " …(i)`
and Re `(z) = x`
Also, `z = x + iy`
`rArr, z^(2) = x^(2) + 2ixy - y^(2)`
`rArr z^(2) = (x^(2) - y^(2)) + 2ixy`
`rArr Re(z^(2) = x^(2) - Y^(2)" " [:. Re (z^(2)) = 0 ]`
`rArr x^(2) - Y^(2) = 0 " " ...(ii)`
From Eqs. (i) and (ii),
`x^(2) + x^(2) = 4`
`rArr 2x^(2) = 4 rArr x^(2) = 2 `
`rArr x= pm sqrt (2)`
`:. " " y =pmsqrt(2)`
`z = x + iy`
`rArr z = sqrt(2)pmisqrt(2),- sqrt(2)pm isqrt(2)`


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