Solve the problem.
-Find an equation for the level curve of the function $\mathrm { f } ( \mathrm { x } , \mathrm { y } ) = \sqrt { \mathrm { y } ^ { 2 } - 16 }$ that passes through the point $( 0,4 )$.
A) $y = 4$
B) $y = - 16$
C) $y = 16$
D) $\mathrm { y } = \pm 4$

Question

Quizplus · Accepted Answer

The level curve of a function $f(x, y)$ at a point $(x_0, y_0)$ is found by setting $f(x, y) = f(x_0, y_0)$. Substituting $(0, 4)$ into $f(x, y) = \sqrt{y^2 - 16}$, we get $f(0, 4) = \sqrt{4^2 - 16} = \sqrt{0} = 0$. Thus, the level curve equation is obtained by setting the function equal to 0: $\sqrt{y^2 - 16} = 0$, which simplifies to $y^2 = 16$. Since the point $(0, 4)$ must satisfy the level curve, and $4$ is a positive root of $y^2 = 16$, the correct equation that describes the level curve passing through $(0, 4)$ is $y = 4$, not $y = \pm 4$, because we are looking for the specific curve that passes through the given point.

Solve the Problem f(x,y)=y2−16\mathrm { f } ( \mathrm { x } , \mathrm { y } ) = \sqrt { \mathrm { y } ^ { 2 } - 16 }f(x,y)=y2−16​

Solve the Problem $\mathrm { f } ( \mathrm { x } , \mathrm { y } ) = \sqrt { \mathrm { y } ^ { 2 } - 16 }$