GED Mathematical Reasoning TestAlgebraic Problem Solving with Graphs and FunctionsHard
A satellite's orbit around Earth can be approximated by an elliptical path. On a coordinate plane, with Earth at the origin (0,0), the satellite's position at a certain time can be given by the coordinates (x, y). If the satellite's path is modeled by the equation x^2/a^2 + y^2/b^2 = 1, what does the variable 'a' represent in this context?
- AThe period of the satellite's orbit.
- BThe radius of a circular orbit.
- CHalf the length of the major axis of the elliptical orbit.
- DThe height of the satellite above Earth.
Show answer & explanationAnswer & explanation
Correct answer: C. Half the length of the major axis of the elliptical orbit.
The equation x^2/a^2 + y^2/b^2 = 1 is the standard form for an ellipse centered at the origin. In this form, 'a' represents the semi-major axis (half the length of the longest diameter of the ellipse), and 'b' represents the semi-minor axis (half the length of the shortest diameter).
Why the other options are wrong
- A. The period is the time it takes for one full orbit, which is not directly represented by 'a' in this equation.
- B. This equation is for an ellipse, not a circle (unless a=b). 'a' is specifically for the ellipse's dimensions.
- D. Height is a single distance, not a parameter of the overall orbital shape.
Standard Equation of an Ellipse
The equation (x^2/a^2) + (y^2/b^2) = 1 describes an ellipse centered at the origin, where 'a' and 'b' are the lengths of the semi-major and semi-minor axes.
- a is semi-major axis (half longest diameter)
- b is semi-minor axis (half shortest diameter)
- If a = b, it's a circle (radius a)
- Foci are on the major axis
Memory trick: Ellipse is Squished Circle: A and B Define!