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 $\frac{\alpha }{\beta }=\frac{\alpha }{\beta }$ $\Rightarrow$ $\frac{\alpha +\beta }{\alpha -\beta }=\frac{\alpha +\beta }{\alpha -\beta }$ $\Rightarrow$ $\frac{{(\alpha +\beta )}^2}{{(\alpha -\beta )}^2}=\frac{{(\alpha +\beta )}^2}{{(\alpha -\beta )}^2}$ $\Rightarrow$ $\frac{b^2}{b^2-4c}=\frac{q^2}{q^2-4r}$ $\Rightarrow$ $b^{2}r=q^{2}c$