Answer
$(0.909,-0.416,2)$ and $(-0.909,-0.416,-2)$
Work Step by Step
The parametric equation from $r(t)= (\sin^2 t+\cos^2 t)+t^2=5$ is given as:
$\sin^2 t+\cos^2 t=1$
We need to plug these parametric equations in to the given paraboloid equation as: $z=x^2+y^2$
$1+t^2=5$
or, $t^2=4$
or, $ t= \pm 2$
Now, the value of $t$ for the points of intersection are:
$ \lt \sin 2, \cos 2, 2 \gt$
and
$ \lt \sin (-2), \cos (-2), 2 \gt$
Thus, the the points of intersection are: $(0.909,-0.416,2)$ and $(-0.909,-0.416,-2)$