Question

If $x^2+ax+10=0$ and $x^2+bx-10=0$ have common root, $a^2-b^2$ is equal to:

• A. $10$
• B. $20$
• C. $30$
• D. $40$

Answer 1

The correct option is D $40$

Let the roots of $x^2+ax+10=0$ be $\alpha$ and $\beta$

$\therefore \alpha +\beta =-a$ ...(1)

and $\alpha \beta =10$ ...(2)

Let the roots of $x^2+bx-10=0$ be $\alpha$ and $\gamma$

$\therefore \alpha +\gamma =-b$ ...(3)

and $\alpha \gamma =-10$ ...(4)

Dividing equation (3) and (4), we get

$\frac{\alpha \beta }{\alpha \gamma }=\frac{10}{-10}$

$\therefore \beta =-\gamma$ ...(5)

$\therefore a^2-b^2=(\alpha +\beta {)}^2-(\alpha +\gamma )$ ...(from 1 and 3)

$={\alpha }^2+{\beta }^2+2\alpha \beta -{\alpha }^2-{\gamma }^2-2\alpha \gamma$

$=({\beta }^2-{\gamma }^2)+2\alpha \beta -2\alpha \gamma$

$=0+20-(-20)$ ...(from 2,4, and 5)

$=40$