Question

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

Answer 1

`\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}`