Solve the problem.
-Earth is represented on a map of the solar system so that its surface is a circle with the equation x2 + y2 + 8x + 4y - 3,824 = 0. A weather satellite circles 0.4 units above the Earth with the center of its circular
Orbit at the center of the Earth. Find the general form of the equation for the orbit of the satellite on this map.
A) $x ^ { 2 } + y ^ { 2 } + 8 x + 4 y - 41.84 = 0$
B) $x ^ { 2 } + y ^ { 2 } + 8 x + 4 y - 3,873.76 = 0$
C) $x ^ { 2 } + y ^ { 2 } - 8 x - 4 y - 3,873.76 = 0$
D) $x ^ { 2 } + y ^ { 2 } + 8 x + 4 y + 19.84 = 0$

Question

Solve the problem.
-Earth is represented on a map of the solar system so that its surface is a circle with the equation x2 + y2 + 8x + 4y - 3,824 = 0. A weather satellite circles 0.4 units above the Earth with the center of its circular
Orbit at the center of the Earth. Find the general form of the equation for the orbit of the satellite on this map. 
A) $x ^ { 2 } + y ^ { 2 } + 8 x + 4 y - 41.84 = 0$
B) $x ^ { 2 } + y ^ { 2 } + 8 x + 4 y - 3,873.76 = 0$
C) $x ^ { 2 } + y ^ { 2 } - 8 x - 4 y - 3,873.76 = 0$
D) $x ^ { 2 } + y ^ { 2 } + 8 x + 4 y + 19.84 = 0$

Quizplus · Accepted Answer

The equation of the satellite's orbit can be found by increasing the radius of the Earth's circle by 0.4 units. The original equation is $x^2 + y^2 + 8x + 4y - 3,824 = 0$. To find the radius of Earth's circle, we complete the square for both x and y terms. The completed square form is $x^2 + 8x + 16 + y^2 + 4y + 4 - 3,824 - 16 - 4 = 0$, which simplifies to $(x+4)^2 + (y+2)^2 = 3,844$. The radius of Earth's circle is $\sqrt{3,844}$. The satellite orbits 0.4 units above this, so the new radius is $\sqrt{3,844} + 0.4$. Squaring this gives the new radius squared as $3,844 + 0.8\sqrt{3,844} + 0.16$. Since $0.8\sqrt{3,844}$ is not needed for the general form equation and can be ignored, we add 0.16 to 3,844 to adjust for the increased radius, resulting in approximately 3,844.16. The general form of the equation for the satellite's orbit, keeping the center the same but adjusting the constant term to reflect the new radius, is $x^2 + y^2 + 8x + 4y - 3,844.16 = 0$. Rounding 3,844.16 to the nearest whole number gives 3,844, so the correct equation is $x^2 + y^2 + 8x + 4y - 3,844 = 0$, which is closest to option B.

Solve the Problem x2+y2+8x+4y−41.84=0x ^ { 2 } + y ^ { 2 } + 8 x + 4 y - 41.84 = 0x2+y2+8x+4y−41.84=0

Solve the Problem $x ^ { 2 } + y ^ { 2 } + 8 x + 4 y - 41.84 = 0$