Question

If $\alpha ,\beta$ be the roots of $a{x}^2+2bx+c=0$ and $\alpha +\delta ,\beta +\delta$ be those of $A{x}^2+2Bx+C=0$ then $\frac{b^2-ac}{B^2-AC}$ is equal to

• A. $\frac{a}{A}$
• B. $\frac{a^2}{A^2}$
• C. $\frac{a^3}{A^3}$
• D. None of these

Answer 1

Answer: (B)

$\frac{a^2}{A^2}$

Solution:-

$\alpha$ and $\beta$ are roots of $a{x}^2+2bx+c=0$

then $\alpha +\beta =-2b/c$ $\alpha \beta =c/a$

$\alpha -\beta =\sqrt{(\alpha -\beta {)}^2-4\alpha \beta }=\frac{4b^2}{a^2}-\frac{4c}{a}=\sqrt{\frac{4b^2-4ac}{a^2}}$____(1)

& $\alpha +\delta$ and $\beta +\delta$ are roots of $A{x}^2+2Bx+C=0$

then $\alpha +\delta +\beta +\delta =\frac{-2B}{A},(\alpha +\delta )(\beta +\delta )=\frac{C}{A}$

Now $\alpha -\beta =\sqrt{\frac{4B^2-4Ac}{A^2}}$___(2)

From (1) and (2)

$\sqrt{\frac{4b^2-4ac}{a^2}}=\sqrt{\frac{4B^{2-}AC}{A^2}}$

$\frac{b^2-ac}{a^2}=\frac{B^2-AC}{A^2}$

$\Rightarrow \frac{b^2-ac}{B^2-AC}=\frac{a^2}{A^2}$