I am given the elliptic curve with homogeneous equation $E: X^3+Y^3+Z^3=0$ over a field $K$ with a point $P_{\infty}=[1:-1:0]$. I am told that you can turn $E(K)$ into a group with identity element $P_{\infty}$ and I am asked to derive a formula for the addition of two points $P=(x_1,y_1)$ and $Q=(x_2,y_2)$ and then I am asked to find the Weierstrass normal form of this curve.
I think that $P+Q = (-x_1-x_2-3\lambda^2b,-\lambda x_3-b)$ where $\lambda = \dfrac{y_2-y_1}{x_2-x_1}$ and $b=y_1-\lambda x_1$ and $x_3 = -x_1-x_2-3\lambda^2b$
(I basically did this using the usual addition in elliptic curves but I am not sure if this is correct.)
For the other part, I know that the Weierstrass Normal Form should be $y^2 = x^3+ax+b$ where $a,b \in \mathbb{R}$ but I don't know how to write this curve in such form.