Question

If $x^2-cx+d=0,x^2-ax+b=0$ have one common root and second has equal roots, then $2(b+d)=$

• A. $0$
• B. $ac$
• C. $a+c$
• D. $a-c$

Answer 1

The correct option is B $ac$

$x^2-ax+b=0$ has equal roots

$\Rightarrow a^2=4b⟶\left(1\right)$

Let root be $\alpha$. $\therefore$ $\alpha =a/2$

Now,

$x^2-cx+d=0$ & $x^2-ax+b=0$ have one root in common.

$\therefore$ one of root of $x^2-cx+d=0$ is $\alpha$. Let other is $\beta$.

$\alpha +\beta =c$ & $\alpha \beta =d\Rightarrow \frac{a}{2}.\beta =d\Rightarrow \beta =\frac{2d}{a}$

$\frac{a}{2}+\frac{2d}{a}=c\Rightarrow a^2+4d=2ac\Rightarrow 2(b+d)=ac$ [B]