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

Answer:

(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)