Question

A planet follows an elliptical path described by 256 x squared plus 16 y squared equals 4096.256x2+16y2=4096.  A comet follows the parabolic path y equals x squared minus 16.y=x2−16.  Where might the comet intersect the orbiting​ planet?

Answer 1

To find the intersection points of the comet and the planet's orbits, we need to solve the system of equations formed by their respective path equations.

Step-by-step explanation:

To find where the comet might intersect the orbiting planet, we need to find the common points between the equation of the planet's elliptical path and the equation of the comet's parabolic path.

By substituting the expression for y in the equation of the planet into the equation of the comet, we get 256x^2 + 16(x^2 - 16) = 4096.

Simplifying this equation and solving for x will give us the x-coordinates where the comet may intersect the planet's orbit. Substituting these x-values back into either the planet's or comet's equation will give us the corresponding y-coordinates.

Answer 2

(0, -16); (4, 0); (-4, 0)

Step-by-step explanation:

planet path is described by: 256x² + 16y² = 4096

comet path is described by: y = x² - 16

When the comet intersect the orbiting​ planet both will have the same (x,y) coordinates. Replacing one path into the other one, we get:

256x² + 16(x² - 16)² = 4096

256x² + 16(x^4 - 32x² + 256) = 4096

256x² + 16x^4 - 512x² + 4096 = 4096

16x^4 - 256x² = 0

x²(16x² - 256) = 0

x = 0 (double root)

or

16x² - 256 = 0

x² = 256/16

x² = 16

x = 4

or

x = -4

Then:

y = 4² - 16 = 0

y = (-4)² - 16 = 0

y = 0 - 16 = -16

So, the points are: (0, -16); (4, 0); (-4, 0)