Identify the trace of the surface defined by when .
A) Circle
B) Ellipse
C) Parabola
D) Hyperbola
E) Line

Question

Identify the trace of the surface defined by  when  .
A) Circle 
B) Ellipse 
C) Parabola 
D) Hyperbola 
E) Line

Quizplus · Accepted Answer

The equation given is in the form of a quadratic equation, which represents a conic section. When the coefficients of the quadratic equation change, the type of the conic section changes. When comparing the equations given, the only difference is in one of the coefficients (the 3rd one). This means that the conic section defined by the first equation is transformed into the conic section defined by the second equation by changing its shape (stretching, shrinking and/or rotating) along its axes. Since the equation given is in the general form of an ellipse (with non-equal coefficients for x^2 and y^2), the traced surface will also be an ellipse.

Identify the Trace of the Surface Defined by When