Question

If k is the ratio of the roots of polynomial $x^2+px-q$, the value of $\frac{k}{1+k^2}$ is:

[1 mark]

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

Answer 1

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

Let ⍺ and ꞵ be the roots.
$k=\frac{\alpha }{\beta }\frac{k}{1+k^2}=\frac{\frac{\alpha }{\beta }}{1+(\frac{\alpha }{\beta }{)}^2}=\frac{\alpha \beta }{{\alpha }^2+{\beta }^2}=\frac{\alpha \beta }{(\alpha +\beta {)}^2-2\alpha \beta }{x}^2+px-q=a{x}^2+bx+c$
⍺ꞵ = Product of roots = $\frac{c}{a}=-q$
⍺ + ꞵ = sum of roots = $\frac{-b}{a}=-p$
$\frac{k}{1+k^2}=\frac{-q}{(-p{)}^2-2q}=\frac{-q}{p^2-2q}$