Question 1

If K is the ratio of the roots of the polynomial x²+px-q, the value of k/1+k² is:

Question 2

$\large\underline{\sf{Solution-}}$

Given Polynomial:

$\rm \longmapsto f(x) = {x}^(2) + px - q$

Let α and β be the zeros of f(x)

Then:

$\rm \longmapsto \alpha + \beta = - p$

$\rm \longmapsto \alpha \beta = - q$

Now, it's given that:

$\rm \longmapsto k = ( \alpha )/( \beta )$

Consider the expression given:

$\rm = \frac{k}{1 + {k}^(2) }$

$\rm = \frac{ ( \alpha )/( \beta ) }{1 + { \frac{ \alpha {}^(2) }{ \beta {}^(2) } }}$

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

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

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

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

$\rm = \frac{ - q }{ {p}^(2) + 2q}$

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