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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

seo-qna
Last updated date: 20th Jun 2024
Total views: 405k
Views today: 5.05k
Answer
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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.

Complete step-by-step 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.