Question

If $\alpha$ and $\beta$ are roots of quadratic equation $x^2-1=0$, then find the equation whose roots are $\frac{2\alpha }{\beta }$ and $\frac{2\beta }{\alpha }$

• A. $x^2+4x+4$
• B. $x^2+4x-4$
• C. $x^2-4x+4$
• D. $-x^2+4x+4$

Answer 1

The correct option is A $x^2+4x+4$

We have,
$x^2-1=0$
So,
$\alpha +\beta =0$.......(i)
$\alpha \times \beta =-1$........(ii)
From (i) and (ii)
$\alpha =-1,\beta =1or\alpha =1,\beta =-1$
So, $\frac{2\alpha }{\beta }=-2,\frac{2\beta }{\alpha }=-2$
S = $\frac{2\alpha }{\beta }+\frac{2\beta }{\alpha }$ = -4
P= $\frac{2\alpha }{\beta }\times \frac{2\beta }{\alpha }$=4
So, the new equation becomes
$x^2-Sx+P=0$
$x^2-(-4)x+4=0$
$\Rightarrow x^2+4x+4=0$
So, option a is correct.