Q.51 - Precalculus Enhanced with Graphing Utilities

Question

 Weather Satellites 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.

Step-by-Step Solution

Answer

The equation for the orbit of the given satellite on this map is  ${(x+1)}^2+{(y+2)}^2=64.6^2$

Step 1.Given information

Given that Weather Satellites Earth is represented on a map of a portion of the solar system so that its surface is the circle with equation $x^2+y^2+2x+4y-4091=0$

and the weather satellite circles 0.6 unit above Earth with the center of its circular orbit at the center of Earth.

We have to  Find the equation for the orbit of the satellite on this map. 

Step 2. Rewriting the equation

Here circle with the equation is $x^2+y^2+2x+4y-4091=0$

First we have to group the  expression containing x and same thing do for y , then we get 

$(x^2+2x)++(y^2+4y)=4091$

 and complete the square of each expression in parentheses.  Then the expression becomes$$\begin{gathered} (x^2+2x+1)+(y^2+4y+4)=4091+5 \sin ce number added on the left side of the equation must be added on the right. \\ {(x+1)}^2+{(y+2)}^2={64}^2 \end{gathered}$$ number added on the left side of the equation must be added on the right.

Step 3. Find center and radius

Here rewritten equation is in the form   ${(x+1)}^2+{(y+2)}^2={64}^2$, this is in the form of the standard form of the equation of circle, then the center of the earth is (-1,-2). and radius is 64. Since weather satellite  circles with the same center but radius is 0.6 grater than the radius of the earth, then the radius of weather satellite is $64+0.6=64.6 and center is ( -1,-2)$

Step 4. Finding the equation for the orbit of the satellite

 According to the given information center is( -1,-2 ) and radius is 64.6.Then equation for the orbit of the satellite is ${(x+1)}^2+{(y+2)}^2=64.6^2$