an experimental satelite is placed into a polar ellipical orbit above the earth. both its apogee and perigee are above the equator and the earths centre is the focus of the ellipical orbit. the earths radius is taken ar 6000km. the furthest point of the satellites orbit is 10000km from the earths surface and the nearest approach is 2000km. determine the height of the satellite above each pole.

ignoring all those pesky zeros, we can set up the ellipse as follows:

center of earth at (0,0)
perigee at (-8,0)
apogee at (16,0)
center at (4,0)

so, the equation is

(x-4)^2/a^2 + y^2/b^2 = 1

plugging in our points, we have
144/a^2 + 0 = 1
a = 12
a^2-c^2 = b^2
b^2 = 12^-4^2 = 128

so our equation is

(x-4)^2/144 + y^2/128 = 1

Now find the polar altitude (when x=0) ...

To determine the height of the satellite above each pole, we need to find the distances between the satellite and the poles.

First, let's find the distance from the satellite to the center of the Earth. We can use the fact that the apogee (furthest point) and perigee (nearest approach) of the satellite's orbit are given relative to the Earth's surface.

Apogee distance = 10000 km
Perigee distance = 2000 km
Earth's radius = 6000 km

To find the distance from the satellite to the center of the Earth, we add the apogee distance to the Earth's radius:
Distance from satellite to center = Apogee distance + Earth's radius = 10000 km + 6000 km = 16000 km.

Next, we find the distance from the Earth's center to each pole using the Pythagorean theorem. The satellite's orbital path is an ellipse, but since its focus is at the center of the Earth, we can consider it as a circle.

Let's call the distance from the center of the Earth to each pole as "h".
Using the Pythagorean theorem, we have:
(Length of the semi-major axis)^2 = (Distance from satellite to center)^2 + (Distance from center to pole)^2

Since the satellite's orbit is above the equator, the distance from the center of the Earth to the poles is the same as the Earth's radius, which is 6000 km. Hence, we have:
(Length of the semi-major axis)^2 = (16000 km)^2 + (6000 km)^2

Now we can solve for the length of the semi-major axis:
(Length of the semi-major axis)^2 = (256,000,000 km^2) + (36,000,000 km^2)
(Length of the semi-major axis)^2 = 292,000,000 km^2
Length of the semi-major axis = √(292,000,000 km^2)

Finally, to find the height above each pole, we subtract the Earth's radius from the length of the semi-major axis:
Height above each pole = Length of the semi-major axis - Earth's radius
Height above each pole = √(292,000,000 km^2) - 6000 km

Calculating this, we get:
Height above each pole ≈ 16639 km

Therefore, the height of the satellite above each pole is approximately 16639 km.