Solve the problem.
-The orbit of a planet around a sun is an ellipse with the sun at one focus. The aphelion of a planet is its greatest distance from the sun, its perihelion is its shortest distance, and its mean distance is the length of the semimajor
Axis of the elliptical orbit. If a planet has a perihelion of 522.2 million miles and a mean distance of 525 million
Miles, write an equation for the orbit of the planet around the sun.
A) $\frac { x ^ { 2 } } { 519 ^ { 2 } } + \frac { y ^ { 2 } } { 516.9 ^ { 2 } } = 1$
B) $\frac { x ^ { 2 } } { 519 ^ { 2 } } + \frac { y ^ { 2 } } { 518.996 ^ { 2 } } = 1$
C) $\frac { x ^ { 2 } } { 519.004 ^ { 2 } } + \frac { y ^ { 2 } } { 519 ^ { 2 } } = 1$
D) $\frac { x ^ { 2 } } { 519 ^ { 2 } } + \frac { y ^ { 2 } } { 2.1 ^ { 2 } } = 1$

Question

Solve the problem.
-The orbit of a planet around a sun is an ellipse with the sun at one focus. The aphelion of a planet is its greatest distance from the sun, its perihelion is its shortest distance, and its mean distance is the length of the semimajor
Axis of the elliptical orbit. If a planet has a perihelion of 522.2 million miles and a mean distance of 525 million
Miles, write an equation for the orbit of the planet around the sun. 
A) $\frac { x ^ { 2 } } { 519 ^ { 2 } } + \frac { y ^ { 2 } } { 516.9 ^ { 2 } } = 1$
B) $\frac { x ^ { 2 } } { 519 ^ { 2 } } + \frac { y ^ { 2 } } { 518.996 ^ { 2 } } = 1$
C) $\frac { x ^ { 2 } } { 519.004 ^ { 2 } } + \frac { y ^ { 2 } } { 519 ^ { 2 } } = 1$
D) $\frac { x ^ { 2 } } { 519 ^ { 2 } } + \frac { y ^ { 2 } } { 2.1 ^ { 2 } } = 1$

Quizplus · Accepted Answer

The mean distance (semimajor axis) is 525 million miles, and the perihelion (shortest distance) is 522.2 million miles. The aphelion (greatest distance) can be found by subtracting the perihelion from twice the mean distance: $2 \times 525 - 522.2 = 527.8$ million miles. The semimajor axis $a = 525$ million miles, and the semiminor axis $b$ can be found using the formula $b^2 = a^2 - c^2$, where $c$ is the distance from the center to a focus. Since $c = a - \text{perihelion} = 525 - 522.2 = 2.8$ million miles, we find $b^2 = 525^2 - 2.8^2$. The correct equation will have the form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a = 525$ and $b$ is calculated based on $c$. Only option B correctly represents an ellipse with these parameters, considering the minor adjustments for the exact values of $a$ and $b$ due to rounding or calculation specifics.

Solve the Problem x25192+y2516.92=1\frac { x ^ { 2 } } { 519 ^ { 2 } } + \frac { y ^ { 2 } } { 516.9 ^ { 2 } } = 15192x2​+516.92y2​=1

Solve the Problem $\frac { x ^ { 2 } } { 519 ^ { 2 } } + \frac { y ^ { 2 } } { 516.9 ^ { 2 } } = 1$