Question

For all complex numbers z of the form$1 + i\alpha$, if${z^2}{\text{ }} = {\text{ }}x{\text{ }} + {\text{ }}iy$, then:
A. ${y^2} - 4x + 2 = 0$
B. ${y^2} + 4x - 4 = 0$
C. ${y^2} - 4x + 4 = 0$
D. ${y^2} + 4x + 2 = 0$

Answer 1

It is given that $z = {\text{1}} + i\alpha$. Squaring it and equating it with ${\text{ }}x{\text{ }} + {\text{ }}iy$is to be done by comparing real and imaginary parts.

Complete step-by-step answer:
We are given that a complex number is in the form $z = {\text{1}} + i\alpha$. It is also given that the square of this complex number is ${\text{ }}x{\text{ }} + {\text{ }}iy$. It is shown as ${z^2}{\text{ }} = {\text{ }}x{\text{ }} + {\text{ }}iy$…… (1)
To solve the question first we need to find the square of the complex number.
$z = {\text{1}} + i\alpha$…… (2)
Squaring equation (2) both the sides we have,
${z^2} = {(1 + i\alpha )^2} = ({\text{1}} + i\alpha )({\text{1}} + i\alpha )$
Solving it further we have,
${z^2} = 1 + i\alpha + i\alpha + {(i\alpha )^2}$
${z^2} = 1 + 2i\alpha - {\alpha ^2}$ (${i^2} = - 1$)
Rearranging the terms we get,
${z^2} = (1 - {\alpha ^2}) + (2\alpha )i$……. (3)
Here $1 - {\alpha ^2}$ is the real part and $2\alpha$ is the imaginary part.
From equation (1) we have,
${z^2}{\text{ }} = {\text{ }}x{\text{ }} + {\text{ }}iy$…….. (1)
Now equating both equation (1) and equation (3) we get,
$x{\text{ }} + {\text{ yi}} = (1 - {\alpha ^2}) + (2\alpha )i$
We have to equate the real part and imaginary part separately. Now equating we get,
$x$= $1 - {\alpha ^2}$ ……. (4) And
${\text{y}}$=$2\alpha$….. (5)
Here, the term alpha ($\alpha$) is parameter. A parameter is a changing variable. To get a relation between x and y as required we have to eliminate alpha ($\alpha$) from them.
Now from equation (5) we get,
${\text{y}}$=$2\alpha$
$\alpha$= $\dfrac{{\text{y}}}{2}$….. (6)
Substituting value from equation (6) in equation (4) we have,
$\Rightarrow x$= $1 - {\alpha ^2}$
$\Rightarrow x$= $1 - {(\dfrac{y}{2})^2}$
$\Rightarrow x = 1 - \dfrac{{{y^2}}}{4}$
Multiplying both sides with and rearranging we get,
${y^2} + 4x - 4 = 0$
Therefore, option (c) ${y^2} + 4x - 4 = 0$ is correct.

Note: When equating complex numbers we only need to equate the real part with the real and imaginary part with the imaginary part of the complex number. Complex numbers are the numbers that can be represented in the form of a+ib.