So I will write what did I get and what is my question.
Question:
The height of the island in meters above sea level is:
$h(x,y)$ = $\frac{1000}{x^2+y^2+xy+1}-500$
Calculate the equation for the shoreline. What kind of curve is this?
Answer:
So lets suppose the water level is at $h(x,y)=0$ , whereby the equation of the shoreline is given by:
$h(x,y)=0$
$\frac{1000}{x^2+y^2+xy+1}-500=0$
$\frac{1000}{x^2+y^2+xy+1}=500$
$500\cdot(x^2+y^2+xy+1)=1000$
$x^2+y^2+xy+1=2$
$x^2+y^2+xy=1$
$x^2+xy+y^2=1\qquad$*(equation 1)
So it is implicit ellipse curve.
My question is:
How do I make the ellipse curve graph from *(equation 1)?
I know I need to use partial derivative to get the graph, but I don't know how from *(equation 1)?
What I need is get *(question 1) into this shape in the below:
