The following images are from Garrity's Algebraic Geometry: A Problem Solving Approach. There is a conflicting expression of tangent line to the curve in projective space. Kindly let me know which one is correct:
In the above problem, the tangent at $(a:b:c)$ is given by $ \left( \frac{\partial P}{\partial x} (a,b,c) \right )x + \dots =0 $ while in the below problem, the tangent seem to be given by $ \left( \frac{\partial P}{\partial x} (a,b,c) \right )(x-a) + \dots =0 $
There is no conflict: as for homogeneous $f$, we have $\sum_i x_i\cdot\frac{\partial f}{\partial x_i} = f\cdot \deg(f)$, we have that $\sum_i \frac{\partial f}{\partial x_i}(p)\cdot p_i = \deg(f)\cdot f(p)$. So the second expression is exactly the same as the first.