Find the equation of the plane that is parallel to $z = 2x + y$ and tangent to $ z = x^2+ y^2$
If it's parallel then $ z_{plane} = kz = 2kx + 2ky $.
If it's tangent then $z_{plane} = z_{0} + 2x_{0}(x - x_{0}) + 2y_{0}(y - y_{0})$.
$ a = 2k, b = k, c = -k$.
$a = 2x_{0}k, b = 2y_{0}k, c = k$.
Therefore the equation of this plane is
$z_{plane} = \frac{5}{4} + 2(x - 1) + y - \frac{1}{2}$.
Is this correct?
Yes, you are correct. A plane parallel to the given one is $$z = 2x + y+k$$ with $k\in\mathbb{R}$. A tangent plane to the paraboloid $z=x^2+y^2$ has the form $$z = z_{0} + 2x_{0}(x - x_{0}) + 2y_{0}(y - y_{0})=2x_0 x+2y_0 y-z_0.$$ Hence, $x_0=1$ and $y_0=1/2$ and $z_0=x_0^2+y_0^2=5/4$. Therefore the tangent plane parallel to the given one is $$z =2x+y-5/4.$$