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  • If the equation, has conjugate complex roots and they satisfy <img alt="\left | z+1 \right |=2\sqrt{10

If the equation, x^{2}+bx+45=0(b\epsilon R) has conjugate complex roots and they satisfy \left | z+1 \right |=2\sqrt{10}, then:
 
Option: 1 b^{2}+b=12
Option: 2 b^{2}-b=42
Option: 3 b^{2}-b=30
Option: 4 b^{2}+b=72
 

Answers (1)

best_answer

 

 

Nature of Roots -

Let the quadratic equation is ax2 + bx + c = 0

D is the discriminant of the equation.

 

i) if D < 0, then root are in the form of complex number, 

   If a,b,c ∈ R (real number) then roots will be conjugate of each other, means if p + iq is one of          

   the roots then other root will be p - iq

 

-

 

 

Let  z=\alpha\pm i\beta be roots of the equation

so 2 \alpha=-b \text { and } \alpha^{2}+\beta^{2}=45,(\alpha+1)^{2}+\beta^{2}=40

So, (\alpha+1)^{2}-\alpha^{2}=-5

\begin{array}{l}{\Rightarrow \quad 2 \alpha+1=-5 \quad \Rightarrow \quad 2 \alpha=-6} \\ {\text { so } \mathrm{b}=6}\end{array}\\Hence,\;\;b^2-b=30

Correct Option 3

Posted by

Kuldeep Maurya

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