Question

Earth is represented on a map of a portion of the solar system so that its surface is the circle with equation x2+y2+8x+2y−4883=0. A weather satellite circles 0.4 unit above Earth with the center of its circular orbit at the center of Earth. Find the general form of the equation for the orbit of the satellite on this map.

Answer 1

Answer:

The general form of the equation for the orbit of the satellite is $x^{2}+y^{2}+8\cdot x+2\cdot y = 4939.16$.

Step-by-step explanation:

From statement we have the general equation of the elliptic form of Earth. First we have to transform this formula to the standard form of the equation of the circle. That is:

1) $x^{2}+y^{2}+8\cdot x +2\cdot y -4883=0$ Given

2) $(x^{2}+8\cdot x)+(y^{2}+2\cdot y) = 4883$ Associative, commutative and modulative properties/Compatibility with addition/Existence of additive inverse.

3) $(x^{2}+8\cdot x +16)+(y^{2}+2\cdot y+1)-17=4883$ Associative, commutative and modulative properties/Compatibility with addition/Existence of additive inverse.

4) $(x+4)^{2}+(y+1)^{2} = 4900$ Square perfect trinomial/Result

According the result, the center of Earth is $C(x,y) = (4, 1)$ and the square of the radius of Earth equals 4900. That is: $r = 70$

If satellite circles 0.4 unit above Earth, then the equation of the orbit of the satellite is:

$(x+4)^{2}+(y+1)^{2}=70.4^{2}$

$(x+4)^{2}+(y+1)^{2} = 4956.16$

Now we transform this into its general form:

1) $(x+4)^{2}+(y+1)^{2} = 4956.16$ Given

2) $x^{2}+8\cdot x +16+y^{2}+2\cdot y+1 = 4956.16$ Square perfect trinomial

3) $x^{2}+y^{2}+8\cdot x+2\cdot y = 4939.16$ Associative, commutative and modulative properties/Compatibility with addition/Existence of additive inverse/Result

The general form of the equation for the orbit of the satellite is $x^{2}+y^{2}+8\cdot x+2\cdot y = 4939.16$.