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# Find the complex number z, the least in absolute value which satisfy the given Condition $\left| {z - 2 + 2i} \right|$ =1

Last updated date: 20th Jun 2024
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Hint: First we will substitute z= x + iy and then we’ll find the magnitude of the given complex number and with the help of that we’ll have the value of x and y. Later on doing the differentiation and equating it to zero, we’ll have the value of $\theta$ and finally on putting the value of $\theta$ we’ll have our answer.

In this question we need to find the complex number z,
Which is satisfying the above condition so let say z = x + iy
And hence on putting the value, we have
$\left| {x + iy - 2 + 2i} \right| = 1$
So now in order to remove modulus we have to find the magnitude of the given equation and hence,
$\sqrt {{{(x - 2)}^2}} + \sqrt {{{\left( {y + 2} \right)}^2}} = 1$
And hence on squaring both sides, we have
${{\text{(x - 2)}}^2} + {(y + 2)^2} = 1$
We know that equation of a circle with centre (h, K) and radius ‘r’ is given as:
${\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2}$
And hence we can conclude that the above equation is an equation of a circle.
So, we can write:
${\text{x - 2 = cos}}\theta$
Now on rearranging we have
${\text{x = 2 + cos}}\theta$
Similarly we can say that
${\text{y = - 2 + sin}}\theta$
And hence on putting the value of x and y we have
${\text{z = 2 - 2i + }}\left( {\cos \theta + i\sin \theta } \right)$
And hence ${\left| z \right|^2} = K = {(\cos \theta + 2)^2} + {(\sin \theta - 2)^2}$
And hence on doing the simplification we have,
$\Rightarrow {\text{1 + 8 + 4}}\left( {\cos \theta - \sin \theta } \right)$
$\Rightarrow 9 + 4\left( {\cos \theta - \sin \theta } \right) = f(\theta )$
Now on doing the differentiation and equating it to zero, we have,
$\dfrac{{df(\theta )}}{{d\theta }}{\text{ = 0}}$
$\dfrac{{d(9 + 4\left( {\cos \theta - \sin \theta } \right))}}{{d\theta }}{\text{ = 0 }}$
$\Rightarrow 4\left( { - \sin \theta - \cos \theta } \right) = 0$
$\therefore \cos \theta = - \sin \theta$
And on simplification, we have
$\Rightarrow {\text{tan}}\theta {\text{ = - 1}}$

$\therefore \theta {\text{ = }}\dfrac{{3\pi }}{4}$
And hence on putting the value of $\theta$ , we have
${\text{z = 2}}\left( {1 - i} \right) + \left( {\dfrac{{ - 1 + i}}{{\sqrt 2 }}} \right)$
Now on separating the real and imaginary part we have,
${\text{z = 0 + (1 - i)}}\left[ {2 - \dfrac{1}{{\sqrt 2 }}} \right]$

Note: In this question, you should have in mind that you have to convert the given expression in terms of a single variable function. This is only possible if we use the parametric form of a circle. Once you find the equation in one variable, you should know how to find the maximum or minimum value of a function. The maxima or minima of a function occur at critical points or at the end points.