Question

Let $\alpha$, $\beta$ be the roots of the equation $x^2-px+r=0$ and $\frac{\alpha}{2},2\beta$ be the roots of the equation $x^2-qx+r=0.$ Then, the value of r is

• A. $\frac{2}{9}(p-q)(2q-p)$
• B. $\frac{2}{9}(q-p)(2p-q)$
• C. $\frac{2}{9}(q-2p)(2q-p)$
• D. $\frac{2}{9}(2p-q)(2q-p)$

Answer 1

The Correct Option is D

The equation $x^2-px+r=0$ has roots $\alpha$, $\beta$ and the equation $x^2-qx+r=0$ has roots $\frac{\alpha}{2},2\beta.$
$\Rightarrow\, \, \, \, \, \, \, r=\alpha\beta$ and $ \alpha+\beta=p,$
and $\frac{\alpha}{2}+2\beta=q\Rightarrow\beta=\frac{2q-p}{3}$ and $ \alpha=\frac{2(2p-q)}{3}$
$\Rightarrow\, \, \, \, \, \, \, \alpha\beta=r=\frac{2}{9}(2q-p)(2p-q)$