Question

The moon is a sphere with radius of 959 km. Determine an equation for the ellipse if the distance of the satellite from the surface of the moon varies from 357 km to 710 km.

Answer 1

well, there's an assumption you should already know the equation for an ellipse

Answer 2

I do. Something like x^2/a^2 + y^2/b^2 = 1, correct?

Answer 3

yes, so, \(\bf \cfrac{(x-h)^2}{a^2}+\cfrac{(y-k)^2}{b^2}=1\)

what does "a" stand for?
what does "b" stand for?

Answer 4

The distances from the center of the circle to the vertices, I think.

Answer 5

yes, the "a" is to the vertices, and the "b" distance to the co-vertices
"a" is the bigger denominator, and the fraction with the bigger denominator, or "a"
is where the ellipse is LONGER

or it has it's major axis
whilst "b" is the minor axis

Answer 6

Ah, all right.

Answer 7

is your ellipse LONGER horizontally or vertically?

Answer 8

It is longer vertically.

Answer 9

thus, is LONGER over the y-axis, thus the fraction with the "y" variable, will have to have the bigger denominator, that is, "a" => \(\bf \cfrac{(x-h)^2}{b^2}+\cfrac{(y-k)^2}{a^2}=1\)

Answer 10

Okay. So we need to find a first?

Answer 11

well, \(\text{the surface of the moon varies from }\bf \text{357 km to 710 km.}\)
those are more or less given

Answer 12

I think those numbers were meant to represent the distance of the satellite from the moon at the furthest and nearest points of orbit. The moon itself is a sphere and not an ellipse.

Answer 13

The file I posted shows the diagram.

Answer 14

That's what the orbit path of the satellite looks like, yes. But the moon itself is located in the center, with its radius of 959 km; 710 km is the distance from the moon to the satellite, say, at the topmost or bottom y-intercept, and 357 km is the distance from the moon to the satellite at the x-intercepts.

Answer 15

I see now :(

Answer 16

Yeah, sorry for any lack of clarity. >.>

Answer 17

Do you have any idea how to solve it?

Answer 18

yes

Answer 19

ok, as I saiid, in the picture, the ellipse is LONGER vertically, so the y-axis fraction will have the bigger denominator, or the "a" component
we just grab the values GIVEN and add the radius of the moon
keep in mind that off the choices given,
can't be A, missing the moon radius
can't be B, "x" fraction has the bigger, which means a horizontal ellipse, which isn't
can't be D, "x" fraction has the bigger, which means a horizontal ellipse, which isn't

so 310 + 959 = b
and 710 + 959 = a

Answer 20

Oh, all right. Great! I understand it now. Thank you so much for your help. :)

Answer 21

\(\bf \cfrac{(x-h)^2}{b^2}+\cfrac{(y-k)^2}{a^2}=1 \implies \cfrac{(x-0)^2}{(357+959)^2}+\cfrac{(y-0)^2}{(710+959)^2}=1\)

Answer 22

Thank you.

Answer 23

yw