Question

Identify the level curves of the following functions:

(i) √(x²+y²)

(ii) √(4 - x² + y²)

(iii) x-y

(iv) x/y

Answer 1

A level curve of a function $f(x,y)$ is the curve with the equation $f(x,y)=c$, where $c$ is an arbitrary constant.

(i) The level curves of the function $\sqrt{x^2+y^2}$ can be determined by the equations:

$\sqrt{x^2+y^2}=r,\, r\geq0$

$x^2+y^2=r^2$ - this is the equation of the circle centered at (0,0) and with radius $r$.

Therefore, the level curves of the function $\sqrt{x^2+y^2}$ is a family of circles centered at (0,0).

(ii) The level curves of the function $\sqrt{4 - x^2 + y^2}$ can be defined by the equations:

$\sqrt{4 - x^2 + y^2}=c,\, c\geq 0$

$- x^2 + y^2=c^2-4$

$(y+x)(y-x)=c^2-4$

We obtained the family of hyperbolas with asymptotes $y-x=0$ and $y+x=0$ (corresponding to $c\ne2$), and the curve $y^2-x^2=0$ (corresponding to $c=2$), which is a union of two lines $y-x=0$ and $y+x=0$

(iii) The level curves of the function $x-y$ can be defined by the equations:

$x-y=c$ . These are all straight lines which are parallel to the line $x-y=0$.

(iv) The level curves of the function $x/y$ can be defined by the equations:

$x/y=c$, or $x=cy,\,y\ne0$ - this is a family of open rays converging at point (0,0), excluding two rays with $y=0$.