SOLUTION: Earth is represented on a map of a portion of the solar system so that its surface is the circle with equation x^2 + y^2 + 2x + 4y - 4091 = 0. A weather satellite circles 0.6 unit

Question 1207911: Earth is represented on a map of a portion of the solar system so that its surface is the circle with equation x^2 + y^2 + 2x + 4y - 4091 = 0. A weather satellite circles 0.6 unit above Earth with the center of its circular orbit at the center of Earth. Find the equation for the orbit of the satellite on this map.
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

If you need a refresher on how to complete the square, or why the process works the way it does, check out this page
https://www.mathsisfun.com/algebra/completing-square.html

x^2 + y^2 + 2x + 4y - 4091 = 0
(x^2 + 2x) + (y^2 + 4y) - 4091 = 0
(x^2 + 2x + 1 - 1) + (y^2 + 4y) - 4091 = 0 .... completing the square for the x terms
(x^2 + 2x + 1) + (y^2 + 4y) - 1 - 4091 = 0
(x+1)^2 + (y^2 + 4y) - 1 - 4091 = 0
(x+1)^2 + (y^2 + 4y + 4 - 4) - 1 - 4091 = 0 .... completing the square for the y terms
(x+1)^2 + (y^2 + 4y + 4) - 4 - 1 - 4091 = 0
(x+1)^2 + (y+2)^2 - 4096 = 0
(x+1)^2 + (y+2)^2 = 4096
(x+1)^2 + (y+2)^2 = 64^2

This fits the circle template
(x-h)^2+(y-k)^2 = r^2
where
(h,k) = (-1,-2) = center
r = 64 = radius
I used GeoGebra to verify this.

The radius 64 is bumped up to 64+0.6 = 64.6 when considering the satellite's orbit.

So,
(x+1)^2 + (y+2)^2 = 64^2
becomes
(x+1)^2 + (y+2)^2 = (64.6)^2
(x+1)^2 + (y+2)^2 = 4173.16 represents one possible equation for the satellite's orbit.

If you want you can expand it out to get it into the form Ax^2+By^2+Cx+Dy+E = 0, but I find it's better to keep it as it is.
You'll have to ask your teacher what s/he prefers most.

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