Please i need help quickly thank you

Question

The guardians of the galaxy have to travel in the spaceship Milano from Xander, located on the star map at (-4,-8) to Titan, located at (6,12), to help the avenger stop Thanos. Unfortunately, the fastest route, y=2x, will fly them through an elliptical asteroid field, represented by the equation x^2/16+y^2/4=1. They will travel an average of 0.53 miles per hour, but they must slow down to navigate the asteroid field. While in the field, they will average 0.39 miles per hour. How long will it take them to make the trip?

Answer 1

Here are some hints.

Determine the points of intersection of the line y = 2x with the ellipse. For this discussion let's call the coordinates of the points of intersection (-a, -b) and (a, b).

Now find the distance from (-4,-8) to (-a,-b) using the distance formula. Do the same for (a, b) to (6, 12).

Use the formula D/R = T to find the time spent travelling outside the asteroid field.

Now find the distance from (-a, -b ) to (a, b). Similarly use D/R = T to find the time spent inside the asteroid field. The sum of those two times will be the total time spent getting from Xander to Titan.

See the following to get the idea:

desmos.com/calculator/kopxai4qbv