Question
Question: If \[z+{{z}^{3}}=0\] then which of the following must be true on the complex plane? \[\left( \text...
If z+z3=0 then which of the following must be true on the complex plane?
(a) Re(z)<0
(b) Re(z)=0
(c) Im(z)=0
(d) z4=1
Solution
Hint : To solve the given question, we will first find out what complex numbers are and what their general form is. Then we will take z common out from the above equation. After doing this, we will get the equation of the form a×b=0. So, we will take up two cases in which we will equate a and b to zero separately. After doing this, we will get some common values from both cases. This common value will be the answer to the question given above.
Complete step-by-step answer :
Before we start to solve the given question, we must know what complex numbers are. A complex number is a number that can be expressed in the form of p + iq, where p and q are real numbers and i represents the imaginary unit, satisfying the equation i2=−1. Now, the equation given in the question is
z+z3=0
We will take z common out from the left-hand side of the above equation. Thus, we will get,
z(1+z2)=0
Now, we know that if a×b=0 then either a = 0 or b = 0 or both. Thus, we will form two cases on the basis of this.
Case I: Let z = 0. We know that z can be written as z = x + iy. Thus, we will get the following equation.
x + iy = 0
Now, the above equation will be zero only if both the real and imaginary parts are equal. Thus, x = 0 and y = 0.
Case II: Let z2+1=0. On subtracting 1 from both the sides, we will get,
⇒z2+1−1=0−1
⇒z2=−1
⇒z2=(i)2
⇒z=±i
Now, we know that z = x + iy. Thus, we will get,
⇒(x+iy)=±i
Now, the above equation will satisfy only when the real and imaginary parts are separately equal. This, x = 0 and y=±1.
From both cases, we can conclude that x must be zero so that the equation z+z3=0 exists. Now, x is the real part of z. Thus, Re (z) = 0.
Hence, option (b) is the right answer.
Note : While applying the formula z = x + iy, we have assumed that both x and y belong to the real number. This assumption is necessary because if x and y are some complex functions then we cannot say that x is the real part of a complex number.