Question 4061
please submit one problem at a time.

<I>A planet's orbit follows a path described by 16x^2 + 4y^2 =64.  A comet follows the parabolic path y=x^2 - 4.  Where might the comet intersect the orbiting planet?</I>

That's a nice one.

That they intersect means that there is a moment of time when their x and y positions, respectively, are the same. If they are both in the point (x, y), that means that 

{{{system( 16x^2+4y^2 = 64, y=x^2-4) }}}

Let's use the second formula for y and substitute it into the first equation:

{{{16x^2 + 4(x^2-4)^2 = 64}}}

Use z for x^2:

{{{16z + 4(z-4)^2 = 64}}}

*[invoke explain_simplification "16z + 4(z-4)^2 = 64"]

Solutions z=4, z=0 mean:

x^2=4, x^2 = 0, or

x=-4, 0, 4.

Use formula {{{y=x^2-4}}} to calculate values of y, for every value of x.