Question

The complex number z satisfying$z + \left| z \right| = 1 + 7i$, then the value of ${\left| z
\right|^2}$ equals
(A) 625
(B) 169
(C) 49
(D) 25

Answer 1

Hint: Assume z as the general complex number and substitute it in the given equation and then
compare the real and the imaginary part. Solve the equation to get the desired result.

It is given in the problem that the complex number z satisfying$z + \left| z \right| = 1 + 7i$.
The goal is to find the value of${\left| z \right|^2}$.
Consider the given equation:
$z + \left| z \right| = 1 + 7i$
First assume that the complex number $z$ as:
$z = x + iy$, where $x$ is the real part of the complex number and $y$ is the imaginary part of
the complex number.
Substitute the value of the complex number in the given equation:
$\left( {x + iy} \right) + \left| {\left( {x + iy} \right)} \right| = 1 + 7i$
The modulus of the complex number is given as:
$\left| {x + iy} \right| = \sqrt {{x^2} + {y^2}} $
Then the above equation becomes:
$\left( {x + iy} \right) + \sqrt {{x^2} + {y^2}} = 1 + 7i$
Now, group the real part and the imaginary part of the complex number in the left- hand side:
$\left( {x + \sqrt {{x^2} + {y^2}} } \right) + iy = 1 + 7i$
In the above equation, the real part in the left- hand side is $\left( {x + \sqrt {{x^2} + {y^2}} }
\right)$and the imaginary part of the left- hand side is $y$.
Now, compare the real and the imaginary part of the equation, so we have

$\left( {x + \sqrt {{x^2} + {y^2}} } \right) = 1$ and $y = 7$
Now, solve the equation:
$\left( {x + \sqrt {{x^2} + {y^2}} } \right) = 1$
$\sqrt {{x^2} + {y^2}} = 1 - x$
Squaring each side of the equation:
${\left( {\sqrt {{x^2} + {y^2}} } \right)^2} = {\left( {1 - x} \right)^2}$
${x^2} + {y^2} = 1 + {x^2} - 2x$
${y^2} = 1 - 2x$
Substitute the value $y = 7$ in the above equation:
${\left( 7 \right)^2} = 1 - 2x$
Now, solve the above equation for $x$:
$49 = 1 - 2x$
$2x = 1 - 49$
$2x = - 48$
$x = - 24$
Now, we have the values:
$x = - 24$ and$y = 7$
So, the complex number is given as:
$z = \left( { - 24} \right) + 7i$
We have to find the value of ${\left| z \right|^2}$, so set the value of the calculated complex
number.
\[{\left| z \right|^2} = {\left| { - 24 + 7i} \right|^2}\]
Simplify the modulus of the complex number:
\[{\left| z \right|^2} = {\left( {\sqrt {{{\left( {24} \right)}^2} + {{\left( 7 \right)}^2}} }
\right)^2}\]
\[{\left| z \right|^2} = {\left( {24} \right)^2} + {\left( 7 \right)^2}\]
\[{\left| z \right|^2} = 576 + 49\]
\[{\left| z \right|^2} = 625\]
So, the required value of ${\left| z \right|^2}$is$625$.

Note: We can also express ${\left| z \right|^2}$ as:z, $\overline z$ where z is the complex number and $\overline z $ is the conjugate complex number.
We can also solve the problem, using this property.