Let [x=(m)/(n)] (m, n are co-prime natural numbers) be a solution of the equation [ (2 ^-1…

Question

Let $$x=\frac{m}{n}$$ ($m, n$ are co-prime natural numbers) be a solution of the equation $$\cos \left(2 \sin ^{-1} x\right)=\frac{1}{9}$$ and let $$\alpha, \beta(\alpha >\beta)$$ be the roots of the equation $m x^2-n x-m+ n=0$. Then th...

Accepted Answer

Assume ${\sin }^{-1}x=	heta $

$\begin{gathered} \cos (2	heta )=\frac{1}{9} \ \sin 	heta =\pm \frac{2}{3} \end{gathered}$

as $m$ and $n$ are co-prime natural numbers,

$x=\frac{2}{3}$

i.e. $m=2,n=3$

So, the quadratic equation becomes $2{x}^{2}-3x+1=0$ whose roots are $\alpha =1,\beta =\frac{1}{2}$

$(1,\frac{1}{2})$ lies on $5x+8y=9$

Solutions · 1