Question

If the roots of the equation $a{x}^2+bx+c=0$ are reciprocal of the roots of the equation $p{x}^2+qx+r=0$, then which of the following options is always correct?

• A. $aq=cr$
• B. $aq=bp$
• C. $bp=aq$
• D. $ap=rc$

Answer 1

The correct option is D $ap=rc$

Let $f\left(x\right)=a{x}^2+bx+c=0\cdots \left(1\right)$ has roots $\alpha ,\beta$
$g\left(x\right)=p{x}^2+qx+r=0$ has roots $\frac{1}{\alpha },\frac{1}{\beta }$
Equation whose roots are $\alpha ,\beta$
$g\left(\frac{1}{x}\right)=0\Rightarrow r{x}^2+qx+p=0\cdots \left(2\right)$
Equation $\left(1\right)$ and $\left(2\right)$ has both roots common,
$\frac{a}{r}=\frac{b}{q}=\frac{c}{p}=k$
$\frac{a}{r}=\frac{b}{q}\Rightarrow aq=br$
$\frac{b}{q}=\frac{c}{p}\Rightarrow bp=cq$
$\frac{a}{r}=\frac{c}{p}\Rightarrow ap=rc$