Question

Let $\alpha ,\beta \in R$. If $\alpha ,{\beta }^2$ be the roots of quadratic equation $x^2-px+1=0$ and ${\alpha }^2,\beta$ be the roots of quadratic equation $x^2-qx+8=0$, then the value of ${}^{\prime }{r}^{\prime }$ if $\frac{r}{8}$ be arithmetic mean of $p$ and $q$, is

• A. $\frac{83}{2}$
• B. $83$
• C. $\frac{83}{8}$
• D. $\frac{83}{4}$

Answer 1

The correct option is B $83$

$x^2-px+1=0$
Product of the roots is,
$\alpha {\beta }^2=1\cdots \left(1\right)$
$x^2-qx+8=0$
Product of the roots is,
${\alpha }^2\beta =8\cdots \left(2\right)$
Using equation $\left(1\right)$ and $\left(2\right),$
$\alpha =4\beta =\frac{1}{2}$
Sum of the roots are,
$\alpha +{\beta }^2=p=4+\frac{1}{4}\Rightarrow p=\frac{17}{4}$
${\alpha }^2+\beta =q=16+\frac{1}{2}\Rightarrow q=\frac{33}{2}$
We know that,
$\frac{r}{8}=\frac{p+q}{2}\Rightarrow r=4\times (\frac{17}{4}+\frac{33}{2})=83$