Question

If $\frac{a}{k},a,ak$ are the roots of $x^3-p{x}^2+qx-r=0$, then $a$ is equal to

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

Answer 1

Answer: (D)

$\frac{q}{p}$

As $\frac{a}{k},a,ak$ are roots of $x^3-p{x}^2+qx-r=0$
Gives $S_3=r=a^3\Rightarrow a=r^{\frac{1}{3}}$
Substituting this in equation
we get
${\left(r^{\frac{1}{3}}\right)}^3-p{\left(r^{\frac{1}{3}}\right)}^2+q\left(r^{\frac{1}{3}}\right)-r=0\Rightarrow -p{r}^{\frac{2}{3}}+q{r}^{\frac{1}{3}}=0\Rightarrow \frac{q}{p}=r^{\frac{1}{3}}=a$