Complex Numbers And Quadratic Equations question 634
Question: $ z_1 $ and $ z_2 $ are the roots of $ 3z^{2}+3z+b=0 $ . If $ O(0), $ $ A(z_1), $ $ B(z_2) $ form an equilateral triangle, then the value of b is
Options:
A) $ -1 $
1
0
D) does not exist
Show Answer
Answer:
Correct Answer: B
Solution:
$ z_1+z_2=-1,z_1z_2=\frac{b}{3} $ $ 0^{2}+z_1^{2}+z_2^{2}=0\times z_1+0\times z_2+z_1z_2 $
$ \Rightarrow {{(z_1+z_2)}^{2}}-2z_1z_2=z_1^2+2z_1z_2+z_2^2-2z_1z_2=z_1^2+z_2^2\Rightarrow 1=3z_1z_2=3\frac{b}{3} $
$ \Rightarrow b=1 $
BETA
