I have the following question for an assignment.
"An irreducible quintic curve in the real projective plane $P^2(R)$ is defined by
$F: (X^2-Z^2)^2Y-(Y^2-Z^2)^2X=0$
Verify that the quartic curve defined by the partial derivative $F_Z$ reduces to an irreducible conic and two lines. Find the four real points of intersection of the conic with $F$. Show that $F$ has no singular points on one line, exactly two singular points on the other line, and exactly two more singular points on the conic. Prove all four singular points of F are crunodes, and find the tangents at these points."
I have managed to find my four singular points, namely $[1,1,1], [1,1,-1], [1,-1,1]$ and $[-1,1,1]$.
I am now confused as to how to approach finding the tangents at these points, as the methods I have been shown seem to be inconsistent and specific to the points in the given question. I have tried taking the affine part with respect to all variables and seem not to get myself anywhere.
Thanks for any help,
Luke