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  4. The set of values of α for which the quadratic equation (α+2) x2-2 α x-α=0 has two roots on the number line symmetrically placed about the point 1 is

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Q. The set of values of $\alpha$ for which the quadratic equation $(\alpha+2) x^2-2 \alpha x-\alpha=0$ has two roots on the number line symmetrically placed about the point 1 is

Complex Numbers and Quadratic Equations

Solution:

Let roots of the equation be
$1-k$ and $1+k$, where $k >0$
Then $2=(1-k)+(1+k)=\frac{2 \alpha}{\alpha+2} \Rightarrow 1=\frac{\alpha}{\alpha+2}$
This is not possible for any value of $\alpha$.