Question

Describe geometrically the set of points in the complex plane satisfying Re(z^2) = 4. Your |answer can be a qualitative description.

Answer 1

Geometrically, the set of points in the complex plane satisfying Re(z^2) = 4 represents a hyperbola centered at the origin with branches opening left and right, symmetrical about the x-axis.

Step-by-step explanation:

To describe geometrically the set of points in the complex plane satisfying Re(z^2) = 4, we need to understand the implication of this equation. For a complex number z, we can write it as z = x + yi where x is the real part and y is the imaginary part. The square of z is z^2 = (x + yi)^2 = x^2 - y^2 + 2xyi. The real part of this expression is x^2 - y^2. The condition Re(z^2) = 4 thus translates to the equation x^2 - y^2 = 4, which is a hyperbola in the complex plane. This hyperbola is centered at the origin and opens left and right because the x^2 term is positive and the y^2 term is subtracted, indicating that the hyperbola’s branches are symmetric about the x-axis.

Answer 2

Answer with Step-by-step explanation:

let z= x+iy

Thus we have

`z^2=(x+iy)^2\\\\z^(2)=x^2+(iy)^2+2ixy\\\\z^2=x^2-y^2+2ixy\\\\Re(z^2)=x^2-y^2`

Thus for
`Re(z^2)=4\\\\x^(2)-y^2=4\\\\(x^2)/(2^2)-(y^2)/(2^2)=1.........(i)`

Comparing the above equation with
`(x^2)/(a^2)-(y^2)/(b^2)=1`

The equation i above is equation of a hyperbola.