I'm having trouble with this. What i do is say $\epsilon: y=mx+b$ is the tangent and it meets the circle at $M_1(x_1,y_1)$, i equate the $y$ of the tangent with the circle: $y=\pm \sqrt{1/2-x^2}$ and then the same with the parabola at $M_2(x_2,y_2)$, but i can't reach a result. I've also tried with this version of the tangent to the circle: $xx_1+yy_1=1/2$. By theory we know that a conic section has a tangent at a given point when the discriminant is zero when we equate the two. I'm very confused. I don't know how to solve this. If someone could help i would be very grateful. Thanks in advance.
Edit
Below i add a figure of the graph i made with Matlab
Best Answer
It is useful to make a sketch to see what is going on. Added: Nice picture, you can see that there are two common tangent lines, that are symmetrical about the $x$-axis.
Let $(a,b)$ be the point of tangency to the circle. Then the tangent line has equation $ax+by=1/2$. To find the point(s) of intersection of this tangent line with the parabola, we solve $y^2=\frac{2-4by}{a}$ or equivalently $$ay^2+4by-2=0.$$ For tangency to the parabola, the above equation has a double root, so the discriminant $16b^2+8a$ is $0$.
We now have the system of equations $a^2+b^2=1/2$, $2b^2=-a$. If we eliminate $b^2$, we get $2a^2-a-1=0$. This has the roots $a=1$ (irrelevant) and $a=-\frac{1}{2}$.