Answer
438.3k+ views
Hint: Here we will first put the value of z in$\left| \dfrac{z-2i}{z+2i} \right|=1$. Then multiply the term with the conjugate $x-iy+2i\Rightarrow x-i(y+2)$. Then solve to find the value of $y$.
Complete step-by- step solution:
Given that: $z=x+iy$and equation is $\left| \dfrac{z-2i}{z+2i} \right|=1$
Here, $\left| \dfrac{z-2i}{z+2i} \right|=1$
We put $z=x+iy$in the equation\[\left| \dfrac{z-2i}{z+2i} \right|=1\]
Now, \[\left| \dfrac{x+iy-2i}{x+iy+2i} \right|=1\]
We multiply with the conjugate [$x-iy+2i\Rightarrow x-i(y+2)$]
\[\Rightarrow \left| \dfrac{x+iy-2i}{x+iy+2i} \right|\left| \dfrac{x-iy-2i}{x-iy+2i} \right|=1\]
Using identity [\[\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}\]] and removing\[\left| {} \right|\]- modulus
\[\Rightarrow \sqrt{\dfrac{{{x}^{2}}+{{\left( y-2 \right)}^{2}}}{{{x}^{2}}+{{\left( y+2 \right)}^{2}}}}=1\]
We, apply cross multiplication
$\Rightarrow \sqrt{{{x}^{2}}+{{\left( y-2 \right)}^{2}}}=\sqrt{{{x}^{2}}+{{\left( y+2 \right)}^{2}}}$
Now squaring each term on both the sides of the equation,
$\Rightarrow {{x}^{2}}+{{\left( y-2 \right)}^{2}}={{x}^{2}}+{{\left( y+2 \right)}^{2}}$
Now expand the equation using identity [${{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$]
$\Rightarrow {{x}^{2}}+{{y}^{2}}+4-4y={{x}^{2}}+{{y}^{2}}+4+4y$
Now we solve and find the value of $y$:
$\Rightarrow 8y=0$$\therefore y=0$
As $y$ is $0$, it will lie on the\[x-axis\].
The locus of the point $z=x+iy$satisfying$\left| \dfrac{z-2i}{z+2i} \right|=1$is \[x-axis\]
Hence, option A is the correct answer.
Note: In these types of problems, where the value is assigned and we need to check a defined condition for an equation - First, we need to put the assigned value in equation then use proper formulas to break it in the simplest form and compare the known and unknown terms. Also, for if any point lies on \[x-axis\]then its corresponding value on the \[y-axis\] will be $0$and vice-versa.
When x, y are real numbers and $x+iy=0,\text{ then x=0 and y=0}$ When a, b, c and d are the real numbers and$a+ib=c+id$ then, $a=c\,\text{ and b=d}$. When the sum of the two complex numbers is the real and the product of the two complex numbers is also real then the given complex numbers are conjugate to each other.
Complete step-by- step solution:
Given that: $z=x+iy$and equation is $\left| \dfrac{z-2i}{z+2i} \right|=1$
Here, $\left| \dfrac{z-2i}{z+2i} \right|=1$
We put $z=x+iy$in the equation\[\left| \dfrac{z-2i}{z+2i} \right|=1\]
Now, \[\left| \dfrac{x+iy-2i}{x+iy+2i} \right|=1\]
We multiply with the conjugate [$x-iy+2i\Rightarrow x-i(y+2)$]
\[\Rightarrow \left| \dfrac{x+iy-2i}{x+iy+2i} \right|\left| \dfrac{x-iy-2i}{x-iy+2i} \right|=1\]
Using identity [\[\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}\]] and removing\[\left| {} \right|\]- modulus
\[\Rightarrow \sqrt{\dfrac{{{x}^{2}}+{{\left( y-2 \right)}^{2}}}{{{x}^{2}}+{{\left( y+2 \right)}^{2}}}}=1\]
We, apply cross multiplication
$\Rightarrow \sqrt{{{x}^{2}}+{{\left( y-2 \right)}^{2}}}=\sqrt{{{x}^{2}}+{{\left( y+2 \right)}^{2}}}$
Now squaring each term on both the sides of the equation,
$\Rightarrow {{x}^{2}}+{{\left( y-2 \right)}^{2}}={{x}^{2}}+{{\left( y+2 \right)}^{2}}$
Now expand the equation using identity [${{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$]
$\Rightarrow {{x}^{2}}+{{y}^{2}}+4-4y={{x}^{2}}+{{y}^{2}}+4+4y$
Now we solve and find the value of $y$:
$\Rightarrow 8y=0$$\therefore y=0$
As $y$ is $0$, it will lie on the\[x-axis\].
The locus of the point $z=x+iy$satisfying$\left| \dfrac{z-2i}{z+2i} \right|=1$is \[x-axis\]
Hence, option A is the correct answer.
Note: In these types of problems, where the value is assigned and we need to check a defined condition for an equation - First, we need to put the assigned value in equation then use proper formulas to break it in the simplest form and compare the known and unknown terms. Also, for if any point lies on \[x-axis\]then its corresponding value on the \[y-axis\] will be $0$and vice-versa.
When x, y are real numbers and $x+iy=0,\text{ then x=0 and y=0}$ When a, b, c and d are the real numbers and$a+ib=c+id$ then, $a=c\,\text{ and b=d}$. When the sum of the two complex numbers is the real and the product of the two complex numbers is also real then the given complex numbers are conjugate to each other.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)