In exercise 3.9 of his Algebraic Geometry book, Prof. Harris asks to show that the rational quartic curves $$C_{a,b}=[X^4-aX^3Y,X^3Y-aX^2Y^2,bX^2Y^2-XY^3,bXY^3-Y^4]$$ are projections of a rational normal curve of $\mathbb{P}^4$ from points $p_{a,b}$.
I managed easily to show that indeed the projections of the rational normal curve of $\mathbb{P}^4$ $$[X^4,X^3Y,X^2Y^2,XY^3,Y^4]$$ from the points $$p_{a,b}=(a^2,a,1,b,b^2)$$ to the hyperplane $$\mathbb{P}^3=[Z_0,Z_1,Z_3,Z_4]$$ give rational quartics projectively equivalent to the family $C_{a,b}$, but I failed to find projections of a rational normal curve of $\mathbb{P}^4$ from points parametrized by $a$ and $b$ on hyperplanes that give exactly the quartic curve $C_{a,b}$.
Can somebody help me find them ?