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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

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

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

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

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

answer is B.

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Detailed Solution

Given, k be the ratio of the roots of the equation

x^{2} + px + q = 0 .

We need to find the value of

\frac{k}{1 + k^{2}} .

We know that,
Sum of roots

= \frac{- b}{a}

Product of roots

= \frac{c}{a}

Assume that the roots of the equation be a and b.
Then,

a + b = p

ab = q

\frac{a}{b} = k

Now solving the equation,

\frac{k}{1 + k^{2}} = \frac{\frac{a}{b}}{1 + {(\frac{a}{b})}^{2}}

\Rightarrow \frac{ab}{a^{2} + b^{2}}

\Rightarrow \frac{ab}{{(a + b)}^{2} - 2 ab}

\Rightarrow \frac{q}{p^{2} - 2 q}

So,

\frac{k}{1 + k^{2}} = \frac{q}{p^{2} - 2 q} .

Hence, the correct option is 2.

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