Intersection between 2 surfaces (skärning mellan ytor)

You have found a curve in the yz-plane which can be used to find a parametric representation of the curve $\gamma(t)$. You might find it more convenient to use $x=t$ as the parameter. Adding 3F to G gives you a curve in the xy-plane:

$4x^2-y^2=3$

Now using $x=t$ as the parameter

$y^2=4t^2-3$

$z^2=4-3t^2$

Around the point $(1,1,1)$ this is equal to the parametric representation

$\mathbf{\gamma}(t)=(x(t), y(t), z(t))=(t,\sqrt{4t^2-3}, \sqrt{4-3t^2})$

$\gamma(1)=(1,1,1)$ (On the curve!)

$\dot{\gamma}(t)=\frac{d\gamma}{dt}=\dots$

Using your original approach and one of $y,z$ as the parameter will give you a slightly more complicated expression. The end result will of course be the same. However, I suspect you are supposed to use the Implicit function theorem on this problem.

How did you derive this part
z²=4−3t²

If you isolate $z^2$ in equation $F$ you get

$z^2=x^2-y^2+1$

Now substitute $x^2=t^2$ and $y^2=4x^2-3$.

It's clear now, thank you for your help! : )