Find the equation of a line that is tangent to both parabolas simultaneously Consider the parabolas $y=x^2$ and $y=x^2-2x+2$. How to find the equation of a line that is tangent to both of them at the same time? Please, walk me through the most intuitive solution.

Hints:

*

*Let $(a, a^2)$ and $(b, b^2 - 2b + 2)$ be the two points of tangency on the common tangent line.

*Independently compute the equation of both tangent lines.


You should get:
$$
\begin{cases}
y - a^2 = 2a(x - a) \\
y - (b^2 - 2b + 2) = (2b - 2)(x - b) \\
\end{cases}
\iff
\begin{cases}
y = (2a)x - a^2 \\
y = (2b - 2)x - b^2 + 2 \\
\end{cases}
$$


*

*Compare coefficients to obtain a system of two equations and two unknowns.


You should get:
$$
\begin{align*}
2a &= 2b - 2 \\
-a^2 &= -b^2 + 2 \\
\end{align*}
$$


*

*Solve the system and plug the values back in to obtain the equation of the tangent line.

