1. Weather Satellites

Earth is represented on a map of a portion of the solar system so that its surface is a 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.

Seeking the first-two steps.

2. Re: Weather Satellites

Put the given circle representing the surface of the Earth into the form:

$\displaystyle (x-h)^2+(y-k)^2=r^2$

Then the trajectory of the satellite will be:

$\displaystyle (x-h)^2+(y-k)^2=\left(r+\frac{3}{5}\right)^2$

3. Re: Weather Satellites

Originally Posted by MarkFL
Put the given circle representing the surface of the Earth into the form:

$\displaystyle (x-h)^2+(y-k)^2=r^2$

Then the trajectory of the satellite will be:

$\displaystyle (x-h)^2+(y-k)^2=\left(r+\frac{3}{5}\right)^2$
x^2 + 2x + y^2 + 4y - 4091 = 0

x^2 + 2x + y^2 + 2y = 4091

x^2 +2x + 1 + y^2 + 4y + 4 = 4091 + 1 + 4

(x + 1)^2 + (y + 2)^2 = 4096

(x + 1)^2 + (y + 2)^2 = 64

So, r = 64.

The equation I seek is

(x + 1)^2 + (y + 2)^2 = (64 + 3/5)

4. Re: Weather Satellites

Originally Posted by harpazo
x^2 + 2x + y^2 + 4y - 4091 = 0

x^2 + 2x + y^2 + 2y = 4091

x^2 +2x + 1 + y^2 + 4y + 4 = 4091 + 1 + 4

(x + 1)^2 + (y + 2)^2 = 4096

(x + 1)^2 + (y + 2)^2 = 64 …. this should be 64^2

So, r = 64.

The equation I seek is

(x + 1)^2 + (y + 2)^2 = (64 + 3/5)^2

Now evaluate the RHS.

5. Re: Weather Satellites

Originally Posted by Debsta
Now evaluate the RHS.
RHS = (323/5)^2

RHS = (104,329)/5

Final Answer: (x + 1)^2 + (y + 2)^2 = (104,329)/5

6. Re: Weather Satellites

Originally Posted by harpazo
RHS = (323/5)^2

RHS = (104,329)/5

Final Answer: (x + 1)^2 + (y + 2)^2 = (104,329)/5

RHS should be on 25 not 5.

7. Re: Weather Satellites

Originally Posted by Debsta
RHS should be on 25 not 5.
Yes, you are right. I forgot to square 5.

RHS = (323/5)^2

RHS = (104,329)/25

Final Answer: (x + 1)^2 + (y + 2)^2 = (104,329)/25

Correct?