Problem 53 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. In order to graph an ellipse whose equation contained an \(x y\) -term, I used a rotated coordinate system that placed the ellipse's center at the origin.

Step-by-Step Solution

Answer

Yes, the statement makes sense. Using a rotated coordinate system is a valid approach for graphing an ellipse with an equation containing \(xy\)-term, and placing the ellipse's center at the origin is a common practice when graphing.

Step 1: Understanding Rotated Coordinate Systems

A rotated coordinate system doesn't alter the center of an ellipse, but it can make graphing the ellipse easier by aligning it with the coordinate axes. If an ellipse equation contains an \(xy\)-term, this suggests that the ellipse is tilted or rotated in relation to the standard coordinate system. Using a rotated coordinate system in this case is a valid approach.

Step 2: Analysing the Statement

When analysing the statement stated in the question, it makes sense. The statement is logical and justified. Using a rotated coordinate system is a valid method of graphing such an ellipse, and placing the ellipse's center at the origin is a common practice when graphing in general.

Step 3: Reasoning

Placing the ellipse's center at the origin in the rotated system doesn't change the shape or orientation of the ellipse. It merely changes the perspective one uses to view the ellipse. Therefore, it's a feasible approach, and the statement makes sense.