Question

planet a’s satellite has an elliptical orbit described by
\\(\frac{x^2}{(5028)^2} + \frac{y^2}{(4703)^2} = 1\\).
(all units are in miles.) the coordinates of the center of planet a are (19,0). complete parts a and b below.
(figure not drawn to scale )
a. the perigee of the satellite’s orbit is the point that is nearest planet a’s center. if the radius of planet a is approximately 3944 miles, find the distance of the perigee above planet a’s surface.
the perigee’s distance above planet a’s surface is \\(\square\\) miles.
b. the apogee of the satellite’s orbit is the point that is the greatest distance from planet a’s center. find the distance of the apogee above planet a’s surface.
the apogee’s distance above planet a’s surface is \\(\square\\) miles.

Explanation:

Step1: Identify orbit parameters

Orbit equation: $\frac{x^2}{5028^2} + \frac{y^2}{4703^2} = 1$. So semi-major axis $a=5028$, semi-minor axis $b=4703$. Center of orbit is $(0,0)$ (standard ellipse), center of Planet A is $(19,0)$.

Step2: Calculate perigee distance to planet center

Perigee is nearest to planet center: $a - c - 19$? No, wait: distance from orbit center to planet center is 19. Perigee point on orbit is $(a,0)=(5028,0)$? No, perigee is closest to planet center: distance from planet center $(19,0)$ to perigee point is $(a - 19)$? Wait, orbit center is $(0,0)$, planet center is $(19,0)$. The closest point on ellipse to $(19,0)$ is along x-axis: left end? No, semi-major axis is along x-axis, so farthest point from orbit center is $(a,0)=(5028,0)$, nearest is $(-a,0)=(-5028,0)$? No, wait perigee is nearest to planet center: distance from $(19,0)$ to nearest orbit point: the orbit's rightmost point is $(5028,0)$, leftmost is $(-5028,0)$. Distance from $(19,0)$ to rightmost: $5028 - 19 = 5009$. Distance to leftmost: $19 + 5028 = 5047$. So perigee is rightmost: 5009 miles from planet center.

Step3: Find perigee height above surface

Subtract planet radius: $5009 - 3944 = 1065$.

Step4: Calculate apogee distance to planet center

Apogee is farthest from planet center: leftmost orbit point distance: $19 + 5028 = 5047$.

Step5: Find apogee height above surface

Subtract planet radius: $5047 - 3944 = 1103$.

Answer:

a. 1065
b. 1103