Common tangent lines of two quadratic functions Find all such lines that are tangent to the following curves:
$$y=x^2$$ and $$y=-x^2+2x-2$$
I have been pounding my head against the wall on this. I used the derivatives and assumed that their derivatives must be equal at those tangent point but could not figure out the equations. An explanation will be appreciated.

If $y=x^2$ we have that $\frac {dy}{dx} = 2x$
This gives the gradient of the tangent at the point $(x, y)=(a, a^2)$
If the tangent line is $y=mx + c$ we therefore have $a^2=(2a)\cdot a+c$ whence $c=-a^2$ and the general tangent line to $y=x^2$ is $$y=2ax-a^2$$
If $y=-x^2+2x-2$ we have $\frac {dy}{dx} = -2x+2$, so that the tangent line $y=mx +c$ at $(x, y)=(b, -b^2+2b-2)$ is $-b^2+2b-2=(-2b+2)b+c$ whence $c=b^2-2$ and the general tangent to the parabola is $$y=(-2b+2)x+b^2-2$$
If the two tangents are to be the same line we equate coefficients to give:$$a=1-b$$ and $$a^2=2-b^2$$
