Question

If $\alpha$, $\beta$ are roots of the equation $x^2+px+q=0$, then the equation whose roots are $\frac{q}{\alpha },\frac{q}{\beta }$ will be

• A. $x^2-qx+p=0$
• B. $x^2+px+q=0$
• C. $x^2-px-q=0$
• D. $q{x}^2+px+q=0$

Answer 1

Answer: (B)

$x^2+px+q=0$

Given $\alpha$, $\beta$ are roots of the equation

$x^2+px+q=0$

$\alpha +\beta =-p$

and $\alpha \beta =q$

Now, sum of roots $=\frac{q}{\alpha }+\frac{q}{\beta }$

$=\frac{q(\alpha +\beta )}{\alpha \beta }$

$\Rightarrow$ Sum of roots$=-p$

Product of roots $=\frac{q}{\alpha }\frac{q}{\beta }$

$=\frac{q^2}{\alpha \beta }$

$\Rightarrow$ Product of roots $=q$

So, the required quadratic equation is

$x^2-(-p)x+q=0$

$x^2+px+q=0$.