If the tangent to the parabola ${\mathrm{y}}^2=\mathrm{x}\text{ at a point }(\mathrm{\alpha },\mathrm{\beta }),\mathrm{\beta }>0$ is also a tangent to the ellipse ${\mathrm{x}}^2+2{\mathrm{y}}^2=1$ then $\mathrm{\alpha } = ....$

$(\mathrm{\alpha },\mathrm{\beta })$ lie on y² = x
$\Rightarrow \mathrm{\alpha }={\mathrm{\beta }}^2 ...\left(1\right)$
tangent equation $\mathrm{y}=\frac{\mathrm{x}}{2\mathrm{\beta }}+\frac{\mathrm{\beta }}{2}$   …(2)
It also tangent to the Ellipse then
$\begin{array}{l}\Rightarrow \left({\mathrm{c}}^2={\mathrm{a}}^2{\mathrm{m}}^2+{\mathrm{b}}^2\right)\\ \begin{array}{rl}\Rightarrow \frac{{\mathrm{\beta }}^2}{4}=\frac{1}{4{\mathrm{\beta }}^2}+\frac{1}{2}& \end{array}\end{array}$
$\begin{array}{l}4{\beta }^4-8{\beta }^2-4=0\\ {\beta }^2=1+\sqrt{2}=2.414\end{array}$