Answer
$r(t)=ti+t^2j+4t^2+t^4k$ or, $r(t) =\lt t, t^2, 4t^2+t^4 \gt$
Work Step by Step
Given: $z=4x^2+y^2$ and $y=x^2$
The given equations can be solved as:
$x^2+y^2=1+2y+y^2$
This gives: $y=\dfrac{x^2-1}{2}$
Plug $x=t$ in the above equation
$y=x^2=t^2$
Thus, we have $z=4x^2+y^2$
or, $z=4t^2+(t^2)^2=4t^2+t^4$
Thus, $r(t)=ti+t^2j+4t^2+t^4k$ (in vector form) and $r(t) =\lt t, t^2, 4t^2+t^4 \gt$
