Consider a circle with its center at the origin (O), as shown. Two operations are allowed on the circle: 1. Scale independently along the x and y axes. 2. Rotation in any direction... Consider a circle with its center at the origin (O), as shown. Two operations are allowed on the circle: 1. Scale independently along the x and y axes. 2. Rotation in any direction about the origin. Which figure among the options can be achieved through a combination of these two operations on the given circle?
Understand the Problem
The question is inquiring about transformations applied to a circle centered at the origin using specified operations: scaling independently along the x and y axes and rotating about the origin. It seeks to identify which figure can be obtained through these two operations.
Answer
The resulting figure is an ellipse.
Answer for screen readers
The resulting figure from the operations is an ellipse.
Steps to Solve
- Understand the Operations on the Circle
The circle centered at the origin (0,0) can be transformed by scaling and rotation. Scaling independently along the x and y axes means transforming it to an ellipse, while rotation will affect its orientation.
- Scaling the Circle
The scaling transformation can be represented mathematically. If we scale the radius of the circle $r$ by factors $a$ (for x-axis) and $b$ (for y-axis), the equation becomes: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$ This will shape the circle into an ellipse if $a \neq b$.
- Rotating the Ellipse
When the ellipse created by scaling is then rotated around the origin by an angle $\theta$, the new coordinates can be calculated using the rotation transformation: $$ x' = x \cos(\theta) - y \sin(\theta) $$ $$ y' = x \sin(\theta) + y \cos(\theta) $$ This rotation preserves the ellipse shape but changes its orientation.
- Identifying Possible Shapes
After performing both operations (scaling and rotation), we can conclude that the resultant shape will still be an ellipse. The precise orientation and axis lengths depend on the specific values chosen for $a$, $b$, and the angle of rotation $\theta$.
The resulting figure from the operations is an ellipse.
More Information
The transformations of scaling and rotation applied to a circle result in an ellipse, which retains a continuous, closed shape but may not be symmetrical if scales along the axes differ.
Tips
- Confusing Shapes: A common mistake is assuming the transformed shape is still a circle. Remember, scaling independently will typically produce an ellipse.
- Incorrect Rotation Application: Be cautious when applying rotation formulas; misplacing coordinates can lead to errors in the final orientation.
