Question

\[R\left( {{z^2}} \right) = 1\]is represented by
A) The parabola \[{x^2} + {y^2} = 1\]
B) The hyperbola \[{x^2} - {y^2} = 1\]
C) Parabola or a circle
D) All of the above

Answer 1

Hint: in this question we have to find what shape given equation represent. First, write the given complex number equation as a combination of real and imaginary numbers. Then equate the real part of the complex number with the given value.



Formula Used:Equation of complex number is given by
\[z = x + iy\]
Where
z is a complex number
x represent real part of complex number
iy is a imaginary part of complex number
i is iota
Square of iota is equal to the negative of one



Complete step by step solution:Given: Real part of square of complex number is equal to 1
Now we have \[R\left( {{z^2}} \right) = 1\]
We know that complex number is written as a combination of real and imaginary number.
\[z = x + iy\]
Where
z is a complex number
x represent real part of complex number
iy is a imaginary part of complex number
\[{z^2} = {x^2} - {y^2} + 2ixy\]
It is given in question that\[R\left( {{z^2}} \right) = 1\]
Real part of \[{z^2}\]is equal to \[{x^2} - {y^2}\]
\[{x^2} - {y^2} = 1\]
This equation represents the equation of hyperbola.



Option ‘B’ is correct



Note: Complex number is a number which is a combination of real and imaginary numbers. So in complex number questions, we have to represent the number as a combination of real and its imaginary part. Imaginary part is known as iota. Square of iota is equal to the negative one.
Don’t try to put the value of z in a given equation first try to simplify it after that value of z would be put in the equation.