Complex Numbers And Quadratic Equations question 8
Question: If $ x^{2}+ax+10=0 $ and $ x^{2}+bx-10=0 $ have a common root, then $ a^{2}-b^{2} $ is equal to [Kerala (Engg.) 2002]
Options:
A) 10
B) 20
C) 30
D) 40
Show Answer
Answer:
Correct Answer: D
Solution:
Let a be a common root, then $ {{\alpha }^{2}}+a\alpha +10=0 $ …….(i) and $ {{\alpha }^{2}}+b\alpha -10=0 $ …….(ii) form (i) - (ii), $ (a-b)\alpha +20=0\Rightarrow \alpha =-\frac{20}{a-b} $ Substituting the value of a in (i), we get $ {{( -\frac{20}{a-b} )}^{2}}+a( -\frac{20}{a-b} )+10=0 $
$ \Rightarrow 400-20a(a-b)+10{{(a-b)}^{2}}=0 $
$ \Rightarrow 40-2a^{2}+2ab+a^{2}+b^{2}-2ab=0 $
$ \Rightarrow a^{2}-b^{2}=40 $ .
BETA
