To find: The points where the comet might intersect the

Procedure used:
An Application of a Nonlinear System:
Step 1: Write a system of equations modeling the conditions in the problem.
Step 2: Solve the system and answer the question asked in the problem.
Step 3: Check the proposed solution in the original wording of the problem."
Calculation:
Multiply ${\displaystyle y=x^2-4}$ by 16 and obtain the equation ${\displaystyle 16y=16x^2-64}$.
Add the both equations and obtain the result as follows.
${\displaystyle (16x^2+4y^2)+\left(16y\right)=64+(16x^2-64)}$
${\displaystyle 4y^2+16y=0}$
${\displaystyle y^2+4y=0}$
${\displaystyle y(y+4)=0}$
On further simplification gives,
${\displaystyle y=0}$ or ${\displaystyle y+4=0}$
${\displaystyle y=0}$ or ${\displaystyle y=-4}$
Substitute ${\displaystyle y=0}$ in the equation ${\displaystyle y=x^2-4}$ and obtain the value of x as follows.
${\displaystyle \left(0\right)=x^2-4}$
${\displaystyle x^2=4}$
${\displaystyle x=\pm 2}$
Substitute ${\displaystyle y=-4}$ in the equation ${\displaystyle y=x^2-4}$ and obtain the value of x as follows.
${\displaystyle (-4)=x^2-4}$
${\displaystyle x^2=0}$
${\displaystyle x=0}$
Thus, for ${\displaystyle y=0,x=\pm 2}$ and for ${\displaystyle y=-4,x=0}$.
Thus, the solution set is ${\displaystyle \{(2,0),(-2,0),(0,-4)\}}$
Check the result, by substituting the obtained solutions in the given original equations ${\displaystyle 16x^2+4y^2=64}$ and ${\displaystyle y=x^2-4}$.
Substitute (2.0) in the given system and check.
${\displaystyle 16{\left(2\right)}^2+4{\left(0\right)}^2=64}$
${\displaystyle 64=64}$
${\displaystyle \left(0\right)={\left(2\right)}^2-4}$
${\displaystyle 0=0}$
Substitute (-2,0) in the given system and check.
${\displaystyle 16{(-2)}^2+4{\left(0\right)}^2=64}$
${\displaystyle 64=64}$
${\displaystyle \left(0\right)={(-2)}^2-4}$
${\displaystyle 0=0}$
Substitute (0,-4) in the given system and check.
${\displaystyle 16{\left(0\right)}^2+4{(-4)}^2=64}$
${\displaystyle 64=64}$
${\displaystyle (-4)={\left(0\right)}^2-4}$
${\displaystyle -4=-4}$
Therefore, the points where the comet might intersect the orbiting planet are ${\displaystyle \{(2,0),(-2,0),(0,-4)\}}$.