ACT (Enhanced)MathematicsMedium
A satellite is orbiting Earth. Its orbital path can be modeled by a circle centered at the origin (0,0) on a coordinate plane. If the satellite passes through the point (3, -4), what is the equation of its orbital path?
- Ax² + y² = 5
- Bx² + y² = 25
- Cx² + y² = 7
- Dx² + y² = 49
Show answer & explanationAnswer & explanation
Correct answer: B. x² + y² = 25
The standard equation of a circle centered at the origin is x² + y² = r². The point (3, -4) lies on the circle, so substitute these values for x and y: 3² + (-4)² = r². This gives 9 + 16 = r², so r² = 25. Thus, the equation is x² + y² = 25.
Why the other options are wrong
- A. This would be r=√5, or if r was incorrectly calculated as 5 and then squared incorrectly.
- C. This would be if 3 + 4 was used instead of 3² + (-4)².
- D. This would be r=7, or if 3² + (-4)² was incorrectly calculated as 49.
Equation of a Circle (Origin)
The algebraic representation of a circle centered at the origin (0,0) with a specific radius 'r', given by x² + y² = r².
- All points (x,y) on the circle are equidistant from the origin.
- The distance is the radius 'r'.
- Based on the Pythagorean theorem.
Memory trick: Circle's Center: Point gives radius-squared, plug it in!