Intersection of the strict transform of a curve and exceptional hypersurface in the blow up of the affine plane.

Question

Consider the curve $Y$ to be $(\frac{t^3}{1-t},\frac{t^4}{1-t})$ with $t$ different from 1. I need to parametrize the curve, calling $x$ the first coordinate ad $y$ the second we find that $t^3=x(1-t)$ then $y=tx$. Now I think I have to assume $x$ not 0 and take $t= y/x$. Then the equation of the curve will be $x^4 -yx^3-y^3=0$ Now I have to consider the blow up of $\mathbb{A}^2$ at (0,0) and compute the intersection of the strict transform of $Y$ with the exceptional hypersurface $E$. My argument was: Consider the coordinates of $\mathbb{P}^1$ $(u : v)$ then the strict transform is given by $xv-yu=x^4 -yx^3-y^3=0$. Assume first $u=1$ then the strict tranform here is $x^2-v^3x-v$ and so intersecting with $x=y=0$ we find $v=0$ i.e. the intersection is $(0:0:1:0)$. Working with $v=1$ we find $(0:0:0:1)$. Is my argument correct$ Thank for the help.