If-and-are-the-roots-of-ax-2-bx-c-0-If-1-and-1-are-the-roots-of-a-1-x-2-b-1-x-c-1-0-If-b-1-a-1-k-b-c-then-k-

Question Number 202120 by MATHEMATICSAM last updated on 21/Dec/23

If α and β are the roots of

{a x}^{2} + b x + c = 0 . If \frac{1 - α}{α} and \frac{1 - β}{β} are

the roots of a_{1} x^{2} + b_{1} x + c_{1} = 0 . If

\frac{b_{1}}{a_{1}} = k + \frac{b}{c} then k = ?

Answered by cortano12 last updated on 21/Dec/23

{\begin{cases} {a x}^{2} + b x + c = 0 \Rightarrow α + β = - \frac{b}{a} \\ a_{1} x^{2} + b_{1} x + c_{1} = 0 \Rightarrow \frac{1}{α} + \frac{1}{β} - 2 = - \frac{b_{1}}{a_{1}} \end{cases}

\Rightarrow - \frac{b}{c} - 2 = - \frac{b_{1}}{a_{1}}

\Rightarrow \frac{b}{c} + 2 = \frac{b_{1}}{a_{1}} = k + \frac{b}{c}

∴ k = 2

Answered by Rasheed.Sindhi last updated on 21/Dec/23

\frac{b_{1}}{a_{1}} = k + \frac{b}{c}

k = \frac{b_{1}}{a_{1}} - \frac{b}{c}

= - (- \frac{b_{1}}{a_{1}}) + (- \frac{b}{a} \div \frac{c}{a})

= - (\frac{1 - α}{α} + \frac{1 - β}{β}) + (α + β) \div α β

= - (\frac{β - α β + α - α β}{α β}) + \frac{α + β}{α β}

= \frac{- (α + β) + 2 α β + (α + β)}{α β} = 2