
If \[lo{g_{\sqrt 3 }}\left( {\dfrac{{{{\left| z \right|}^2} - \left| z \right| + 1}}{{2 + |z|}}} \right) < 2\], then the locus of z is
A) \[\left| z \right| = 5\]
B) \[\left| z \right| < 5\]
C) \[\left| z \right| > 5\]
D) None of these
Answer
163.8k+ views
Hint: in this question we have to find the locus of complex number z. Just simplify the given equation and use the factorization method to find the roots of the equation. Then use select the root which satisfy the condition of mod function
Formula Used:Factorization is used to find root of complex number
\[(a + b)(c + d) = ac + ad + bc + bd\]
Where
A,b,c,d are any numbers or variable
Complete step by step solution:Given: A complex number equation
Now we have complex number \[lo{g_{\sqrt 3 }}\left( {\dfrac{{{{\left| z \right|}^2} - \left| z \right| + 1}}{{2 + |z|}}} \right) < 2\]
\[\left( {\dfrac{{{{\left| z \right|}^2} - \left| z \right| + 1}}{{2 + |z|}}} \right) < {(\sqrt 3 )^2}\]
\[|z{|^2} - 4|z| - 5 < 0\]
\[|z{|^2} - 5|z| + |z| - 5 < 0\]
\[|z|(|z| - 5) + (|z| - 5) < 0\]
\[(|z| - 5)(|z| + 1) < 0\]
\[ - 1 < |z| < 5\]
\[|z| < 5\] (Because\[|z| > 0\])
Option ‘B’ is correct
Note: Complex number is a number which is a combination of real and imaginary numbers. So in complex number questions, we have to represent the number as a combination of real and its imaginary part. Imaginary part is known as iota. Square of iota is equal to the negative one.
Don’t try to put the value of z in a given equation first try to simplify it after that value of z would be put in the equation.
Formula Used:Factorization is used to find root of complex number
\[(a + b)(c + d) = ac + ad + bc + bd\]
Where
A,b,c,d are any numbers or variable
Complete step by step solution:Given: A complex number equation
Now we have complex number \[lo{g_{\sqrt 3 }}\left( {\dfrac{{{{\left| z \right|}^2} - \left| z \right| + 1}}{{2 + |z|}}} \right) < 2\]
\[\left( {\dfrac{{{{\left| z \right|}^2} - \left| z \right| + 1}}{{2 + |z|}}} \right) < {(\sqrt 3 )^2}\]
\[|z{|^2} - 4|z| - 5 < 0\]
\[|z{|^2} - 5|z| + |z| - 5 < 0\]
\[|z|(|z| - 5) + (|z| - 5) < 0\]
\[(|z| - 5)(|z| + 1) < 0\]
\[ - 1 < |z| < 5\]
\[|z| < 5\] (Because\[|z| > 0\])
Option ‘B’ is correct
Note: Complex number is a number which is a combination of real and imaginary numbers. So in complex number questions, we have to represent the number as a combination of real and its imaginary part. Imaginary part is known as iota. Square of iota is equal to the negative one.
Don’t try to put the value of z in a given equation first try to simplify it after that value of z would be put in the equation.
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