Question

If \[\left| {{Z^2} - 1} \right| = {\left| Z \right|^2} + 1\] then find the \[Z\] lies on
A. Circle
B. The imaginary axis
C. A real axis
D. An ellipse

Answer 1

Hint: In this question, we need to find the location of \[Z\] where, \[Z\] is a complex number. For this, we need to assume that \[Z = a + ib\]. We can decide the location of \[Z\] based on the value that will come after substituting the value of \[Z\] in the given equation. For this, we will use the concept of modulus of a complex number.

Formula used: The modulus of a complex number \[Z = a + ib\] is \[\sqrt {{a^2} + {b^2}} \]

Complete step-by-step answer: We know that \[\left| {{Z^2} - 1} \right| = {\left| Z \right|^2} + 1\]
Let \[Z = a + ib\]
Now, put \[Z = a + ib\] in the equation \[\left| {{Z^2} - 1} \right| = {\left| Z \right|^2} + 1\]
Thus, we get \[\left| {{{\left( {a + ib} \right)}^2} - 1} \right| = {\left| {a + ib} \right|^2} + 1\]
Thus, the modulus of \[\left| {{{\left( {a + ib} \right)}^2} - 1} \right|\] is given by
\[
   \Rightarrow \left| {{{\left( {a + ib} \right)}^2} - 1} \right| = \left| {\left( {{a^2} + 2abi + {{\left( {ib} \right)}^2}} \right) - 1} \right| \]
We know that \[i^2=-1 \],
\[
   \Rightarrow \left| {{{\left( {a + ib} \right)}^2} - 1} \right| = \left| {\left( {{a^2} + 2abi - {b^2}} \right) - 1} \right| \\
   \Rightarrow \left| {{{\left( {a + ib} \right)}^2} - 1} \right| = \left| {\left( {{a^2} - {b^2} - 1} \right) + 2abi} \right| \\
 \]
We know that the modulus of a complex number \[Z = a + ib\] is \[\sqrt {{a^2} + {b^2}} \]
By applying this concept, we get
\[ \Rightarrow \left| {{{\left( {a + ib} \right)}^2} - 1} \right| = \sqrt {{{\left( {{a^2} - {b^2} - 1} \right)}^2} + {{\left( {2ab} \right)}^2}} \]
By simplifying, we get
\[
   \Rightarrow \left| {{{\left( {a + ib} \right)}^2} - 1} \right| = \sqrt {1 + {a^4} + {b^4} + 4{a^2}{b^2} - 2{a^2}{b^2} - 2{a^2} + 2{b^2}} \\
   \Rightarrow \left| {{{\left( {a + ib} \right)}^2} - 1} \right| = \sqrt {1 + {a^4} + {b^4} + 2{a^2}{b^2} - 2{a^2} + 2{b^2}} \\
 \]
Also, the modulus of \[{\left| {\left( {a + ib} \right)} \right|^2}\] is given by
\[
   \Rightarrow {\left| {\left( {a + ib} \right)} \right|^2} = \left| {a{}^2 + {{\left( {ib} \right)}^2} + 2abi} \right| \\
   \Rightarrow {\left| {\left( {a + ib} \right)} \right|^2} = \left| {\left( {a{}^2 - {b^2}} \right) + 2abi} \right| \\
 \]
Thus, we get
\[
   \Rightarrow \sqrt {{{\left( {{a^2} - {b^2}} \right)}^2} + {{(2ab)}^2}} = \sqrt {{a^4} - 2{a^2}{b^2} + {b^4} + 4{a^2}{b^2}} \\
   \Rightarrow \sqrt {{{\left( {{a^2} - {b^2}} \right)}^2} + {{(2ab)}^2}} = \sqrt {{a^4} + 2{a^2}{b^2} + {b^4}} \\
   \Rightarrow \sqrt {1 + {a^4} + {b^4} + 2{a^2}{b^2} - 2{a^2} + 2{b^2}} = \sqrt {{{\left( {{a^2} + {b^2}} \right)}^2}} \\
   \Rightarrow \sqrt {1 + {a^4} + {b^4} + 2{a^2}{b^2} - 2{a^2} + 2{b^2}} = \left( {{a^2} + {b^2}} \right) \\
 \]
Now, consider \[\left| {{Z^2} - 1} \right| = {\left| Z \right|^2} + 1\]
\[\sqrt {1 + {a^4} + {b^4} + 2{a^2}{b^2} - 2{a^2} + 2{b^2}} = ({a^2} + {b^2} + 1)\]
By taking square on both sides, we get
\[
   \Rightarrow 1 + {a^4} + {b^4} + 2{a^2}{b^2} - 2{a^2} + 2{b^2} = {({a^2} + {b^2} + 1)^2} \\
   \Rightarrow {({a^2} - {b^2} - 1)^2} + 4{a^2}{b^2}\; = {\text{ }}{({a^2} + {b^2} + 1)^2} \\
   \Rightarrow 4{a^2}{b^2}\; = {\text{ }}{({a^2} + {b^2} + 1)^2} - {({a^2} - {b^2} - 1)^2} \\
   \Rightarrow 4{a^2}{b^2}\; = {\text{ }}{b^4} + {a^4} + 2{b^2} + 2{a^2} + {a^2}{b^2} + {b^2}{a^2} + 1 - {b^4} - {a^4} - 2{b^2} + 2{a^2} + {a^2}{b^2} + {b^2}{a^2} - 1 \\
 \]
By simplifying further, we get
\[ \Rightarrow 4{a^2}{b^2}\; = {\text{ }}4{a^2} + 4{a^2}{b^2}\]
\[
   \Rightarrow 4{a^2}{b^2}\; - 4{a^2}{b^2} = {\text{ }}4{a^2} \\
   \Rightarrow {\text{ }}4{a^2} = 0 \\
   \Rightarrow {\text{ }}{a^2} = 0 \\
 \]
By taking square root on both sides, we get
\[ \Rightarrow {\text{ }}a = 0\]
That means the real part is zero.
Hence, this indicates that the \[Z\] lies on the imaginary axis.
Thus, \[Z\] lies on the imaginary axis if \[\left| {{Z^2} - 1} \right| = {\left| Z \right|^2} + 1\].

Therefore, the option (B) is correct.

Additional Information: A complex number is a combination of real and imaginary number. It can be denoted as \[a + ib\] where, \[a\] is a real part and \[ib\] is the imaginary part. The modulus of a complex number is defined as the distance measured from the origin of the point on the argand plane expressing the complex number. The modulus of a complex number can be determined by taking the square root of the sum of the squares of the complex number's real and imaginary parts.

Note: Many students make mistakes in finding modulus. Also, in a complex number if the real part is zero then only the imaginary part exists and if the imaginary part is zero then only the real part exists.