A math/engineering problem?

Question

"Two countries are on a coordinate grid. The US is located at (2,3) and Japan is located at (8,3). Figure out where to put a satellite that will reach both countries. The satellite has a range where the sum of the distances from both countries must be 12 units. Your task is to come up with the equation of the path that the satellite must travel in standard and general form and to determine 5 coordinates where the satellite can be at any particular time." This is a math problem that we were given for homework. Our teacher said it covers a majority of trigonometry and analytical geometry. Please help! I have no idea what to do!! D:

Answer 1

if you consider earth flat then locus of position would be a ellipse with focus2,3 and 8,3 with major axis parallel to x axis
2a=12
a=6
2ae=distance between foci
using e from above find b^2=a^2(1-e^2)
center(h,k) is midpoint of foci
equation:
(x-h)^2/a^2 +(y-k)^2/b^2 =1

Answer 2

This is an ellipse with foci at (2,3) and (8,3).
The equation of an ellipse with major axis parallel to the x-axis is (x-h)^2/a^2 + (y-k)^2/b^2 =1 , where (h,k) are the coordinates of the center of the ellipse, a is the semi major axis and b is the semi minor axis.
The center of the ellipse is halfway between the foci so it is at (5,3). a equals 12 so a^2 = 144 and b^2 = 6^2-3^2= 27
So the equation is (x-5)^2/144 + (y- 3)^2/27 = 1 which is the standard form. You could expand the above equation to get the general form.

Set y = 0 and sove for x getting two coordinates which are (a,0) and (-a,0) Two more points are (0,b) and (0,-b)
Stick in a value of 4 for x and solve for y to get a fifth point.

I never got a problem like this in any of my EE courses.

Answer 3

since at any point of the path, the sum is constant, then the satellite will be travelling in elliptical path with center at (5,3). the location of the 2 countries are the focal points, then the equation should be:

(x-5)^2 / a^2 + (y-3)^2 / b^2 = 1

where by pythagorean theorem, b^2 = (6^2- 3^2) = 27
a=3+(12-6)/2 = 6; a^2 = 36

Answer 4

It assume the satellite orbits a flat earth, right? --Not a round one?