[Elective 4-4: Coordinate Systems and Parametric Equations] (10 points) In the rectangular coordinate system $x O y$, the parametric equation of curve $C _ { 1 }$ is $\left\{ \begin{array} { l } x = \cos ^ { k } t , \\ y = \sin ^ { k } t \end{array} \right.$ ($t$ is the parameter). Establishing a polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of curve $C _ { 2 }$ is $$4 \rho \cos \theta - 16 \rho \sin \theta + 3 = 0$$ (1) When $k = 1$ , what type of curve is $C _ { 1 }$? (2) When $k = 4$ , find the rectangular coordinates of the common points of $C _ { 1 }$ and $C _ { 2 }$ .
[Elective 4-4: Coordinate Systems and Parametric Equations] (10 points)
In the rectangular coordinate system $x O y$, the parametric equation of curve $C _ { 1 }$ is $\left\{ \begin{array} { l } x = \cos ^ { k } t , \\ y = \sin ^ { k } t \end{array} \right.$ ($t$ is the parameter). Establishing a polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of curve $C _ { 2 }$ is
$$4 \rho \cos \theta - 16 \rho \sin \theta + 3 = 0$$
(1) When $k = 1$ , what type of curve is $C _ { 1 }$?
(2) When $k = 4$ , find the rectangular coordinates of the common points of $C _ { 1 }$ and $C _ { 2 }$ .