Describe the level curves of the function $f ( x , y ) = \sqrt { 1 - x ^ { 2 } - 2 y ^ { 2 } }$ .
A)Concentric circles
B)Non-concentric circles
C)Concentric ellipses (not circles)
D)Non-concentric ellipses (not circles)
E)Parabolas with the same vertex
F)Parabolas with the same focus
G)Hyperbolas with the same vertices
H)Hyperbolas with the same foci

Question

Quizplus · Accepted Answer

The level curves of the function $f(x, y) = \sqrt{1 - x^2 - 2y^2}$ are obtained by setting $f(x, y)$ equal to a constant $k$. This gives the equation $1 - x^2 - 2y^2 = k^2$, which can be rearranged to $x^2 + 2y^2 = 1 - k^2$. This equation represents ellipses with their major and minor axes aligned with the coordinate axes, and since the value of $1 - k^2$ simply scales the size of the ellipses without changing their center or orientation, the ellipses are concentric. They are not circles because the coefficients of $x^2$ and $y^2$ are different, indicating that the stretching along the x and y axes are different, thus forming ellipses.

Describe the Level Curves of the Function f(x,y)=1−x2−2y2f ( x , y ) = \sqrt { 1 - x ^ { 2 } - 2 y ^ { 2 } }f(x,y)=1−x2−2y2​

Describe the Level Curves of the Function $f ( x , y ) = \sqrt { 1 - x ^ { 2 } - 2 y ^ { 2 } }$