Question

If α and β be the roots of the equation $2x^2+2(a+b)x+a^2+b^2=0$, then the equation whose roots are $(\alpha +\beta {)}^2$ and $(\alpha -\beta {)}^2)$ is

• A.
• B.
• C.
• D. None of these

Answer 1

The correct option is B

Sum of roots α + β = -(a + b) and αβ = $\frac{a^2+b^2}{2}$

⇒ $(\alpha +\beta {)}^2=(a+b{)}^{2}and(\alpha -\beta {)}^2={\alpha }^2+{\beta }^2-2\alpha \beta$

$=2ab-(a^2+b^2)=-(a-b{)}^2$

Now the required equation whose roots are

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

$x^2-\left\{\right(\alpha +\beta {)}^2+(\alpha -\beta {)}^2\}x+(\alpha +\beta {)}^2(\alpha -\beta {)}^2=0$

⇒ $x^2-\left\{\right(a+b{)}^2+(a-b{)}^2\}x+(a+b{)}^2(a-b{)}^2=0$

⇒ $x^2-4abx-(a^2-b^2{)}^2=0$