Complex Numbers And Quadratic Equations question 470
Question: The difference between the corresponding roots of $ x^{2}+ax+b=0 $ and $ x^{2}+bx+a=0 $ is same and $ a\ne b $ , then
Options:
A) a+b+4=0
B) a+b-4=0
C) a-b-4=0
D) a-b+4=0
Show Answer
Answer:
Correct Answer: A
Solution:
[a] Let $ \alpha ,\beta $ and $ \gamma ,\delta $ be the roots of the equations $ x^{2}+ax+b=0 $ and $ x^{2}+bx+a=0 $ , respectively therefore, $ \alpha +\beta =-a,\alpha \beta =b $ And $ \delta +\gamma =-b,\gamma \delta =a. $ Given $ | \alpha -\beta |=| \gamma -\delta | $
$ \Rightarrow {{(\alpha +\beta )}^{2}}-4\alpha \beta ={{(\gamma +\delta )}^{2}}-4\gamma \delta $
$ \Rightarrow a^{2}-4b=b^{2}-4a $
$ \Rightarrow (a^{2}-b^{2})+4(a-b)=0 $
$ \Rightarrow a+b+4=0 $ $ (\because a\ne b) $
BETA
