SOLUTION: The orbit of the planet has the shape of an ellipse, and on one of the foci is the star around which it revolves. The planet is closest to the star when it is at one vertex. It is

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: The orbit of the planet has the shape of an ellipse, and on one of the foci is the star around which it revolves. The planet is closest to the star when it is at one vertex. It is       Log On


   

Question 1170524: The orbit of the planet has the shape of an ellipse, and on one of the foci is the star around which it revolves. The planet is closest to the star when it is at one vertex. It is farthest from the star when it is at the other vertex. Supposed the closest and farthest distances of the planet from this star, are 420 million kilometers and 580 million kilometers respectively. Find the equation of the ellipse, in standard form, with center at the origin and the star at the x-axis. Assume all units are in millions of kilometers.
Answer by greenestamps(13200) About Me  (Show Source):
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The standard form of the equation of an ellipse is

x%5E2%2Fa%5E2-y%5E2%2Fb%5E2+=+1

a is the length of the semi-major axis; b is the length of the semi-minor axis.

a and b are related by

c%5E2+=+a%5E2-b%5E2

where c is the distance from the center to each focus.

The length of the major axis is the distance between the two extremes of the orbit. In this problem, in units of millions of kilometers, the major axis is 420+580 = 1000. So the semi-major axis a is 500.

With the semi-major axis 500 and the minimum and maximum distances of the planet from the star being 420 and 580, the distance from the center to each focus is 80.

So a=500 and c=80.

(1) Use c%5E2+=+a%5E2-b%5E2 to determine b^2
(2) Write the equation using a^2 and b^2