For all complex numbers z of the form 1 + i α ,   α ∈ R , if z 2 = x + i y , then

Question

For all complex numbers z of the form 1 + i &#945; , &#160; &#945; &#8712; R , if z 2 = x + i y , then

Aicharya · Accepted Answer

Given z = 1 + i&#945; , &#160; z 2 = x + iy &#8658; 1 + i&#945; 2 = x + iy &#8658; 1 2 + i 2 &#945; 2 + 2 i &#945; = x + i y Using i 2 = - 1 , &#8658; 1 - &#945; 2 + 2 i &#945; = x + iy Equating the real and imaginary parts, we get x = 1 - &#945; 2 and y = 2 &#945; &#8658; &#945; 2 = 1 - x and y 2 = 4 &#945; 2 &#8658; y 2 = 4 1 - x &#8658; y 2 = 4 - 4 x &#8658; y 2 + 4 x - 4 = 0 .

For all complex numbers z of the form 1 + i &#945; , &#160; &#945; &#8712; R , if z 2 = x + i y , then

Solution

For all complex numbers z of the form 1 + i α , α ∈ R , if z 2 = x + i y , then