Question

If k be the ratio of the roots of the equation $x^2-px+q=0$ , the value of $\frac{k}{1+k^2}$ is

• A. $\frac{q^2-2p}{p}$
• B. $\frac{q}{p^2-2q}$
• C. $\frac{p}{q^2-2p}$
• D. $\frac{p}{p^2-2q}$

Answer 1

The correct option is B $\frac{q}{p^2-2q}$

Consider the given equation.

$x^2-px+q=0$ $...........\left(1\right)$

Let the roots are $\alpha ,\beta$.

Since, $k=\frac{\alpha }{\beta }$

Since,

$=\frac{k}{1+k^2}$

$=\frac{\frac{\alpha }{\beta }}{1+\frac{{\alpha }^2}{{\beta }^2}}$

$=\frac{\alpha \beta }{{\alpha }^2+{\beta }^2}$

$=\frac{\alpha \beta }{(\alpha +\beta {)}^2-2\alpha \beta }$

$=\frac{q}{p^2-2q}$

Hence, this is the answer.