Complex Numbers And Quadratic Equations question 579
Question: If the ratio of the roots of $ x^{2}+bx+c=0 $ and $ x^{2}+qx+r=0 $ be the same, then [EAMCET 1994]
Options:
A) $ r^{2}c=b^{2}q $
B) $ r^{2}b=c^{2}q $
C) $ rb^{2}=cq^{2} $
D) $ rc^{2}=bq^{2} $
Show Answer
Answer:
Correct Answer: C
Solution:
Let $ \alpha ,\beta $ be the roots of $ x^{2}+bx+c=0 $ and $ \alpha ‘,\beta ’ $ be the roots of $ x^{2}+qx+r=0 $ . Then $ \alpha +\beta =-b,\alpha \beta =c,\alpha ‘+\beta ‘=-q,\alpha ‘\beta ‘=r $ It is given that $ \frac{\alpha }{\beta }=\frac{\alpha ‘}{\beta ‘}\Rightarrow \frac{\alpha +\beta }{\alpha -\beta }=\frac{\alpha ‘+\beta ‘}{\alpha ‘-\beta ‘} $
Þ $ \frac{{{(\alpha +\beta )}^{2}}}{{{(\alpha -\beta )}^{2}}}=\frac{{{(\alpha ‘+\beta ‘)}^{2}}}{{{(\alpha ‘-\beta ‘)}^{2}}}\Rightarrow \frac{b^{2}}{b^{2}-4c}=\frac{q^{2}}{q^{2}-4r} $
Þ $ b^{2}r=q^{2}c $
BETA
