Question

If the ratio of the roots of $ {{x}^{2}}+bx+c=0 $ and $ {{x}^{2}}+qx+r=0 $ is the same, then:

• A. $ {{r}^{2}}b=q{{c}^{2}} $
• B. $ r{{b}^{2}}=c{{q}^{2}} $
• C. $ {{r}^{2}}c=q{{b}^{2}} $
• D. $ r{{c}^{2}}=b{{q}^{2}} $
• E. $ {{r}^{2}}q={{c}^{2}}b $

Answer 1

Answer: (B)

Let $ \alpha ,\beta $ be the roots of $ {{x}^{2}}+bx+c=0 $ and $ \alpha ,\beta $ are 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 } $ $ \Rightarrow $ $ \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} $ $ \Rightarrow $ $ {{b}^{2}}r={{q}^{2}}c $