Question

If $a$ and $b$ are rational and $\alpha ,\beta$ be the roots of $x^2+2ax+b=0,$ then the equation with rational coefficients one of whose roots is $\alpha +\beta +\sqrt{{\alpha }^2+{\beta }^2}$ is

• A. $x^2+4ax-2b=0$
• B. $x^2+4ax+2b=0$
• C. $x^2-4ax+2b=0$
• D. $x^2-4ax-2b=0$

Answer 1

The correct option is B $x^2+4ax+2b=0$

Since $\alpha ,\beta$ are roots of $x^2+2ax+b=0$

$\alpha +\beta =-2a$ and $a\beta =b.$

Let $y=\alpha +\beta +\sqrt{{\alpha }^2+{\beta }^2}$

$\Rightarrow {(y+2a)}^2={\alpha }^2+{\beta }^2={(\alpha +\beta )}^2-2\alpha \beta =4{\alpha }^2-2b$

$\Rightarrow y^2+4ay+2b=0$

So, the required equation is $x^2+4ax+2b=0.$