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 the point $$(\alpha, \beta)$$ lies on the line

Accepted Answer

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

$$\begin{aligned}
& \cos (2 \theta)=\frac{1}{9} \
& \sin \theta= \pm \frac{2}{3}
\end{aligned}$$

as $$\mathrm{m}$$ and $$\mathrm{n}$$ are co-prime natural numbers,

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

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

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

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

Solutions · 1