TITLE: Resemblance of the graph of $\ln (ax^2+by^2)=y^{n}$ with ellipse $ax^2+by^2=1$ for large values of $n$.
QUESTION [6 upvotes]: Coincidentally using Desmos, I found out that the graph (specifically its curvature) of $\ln (x^2+y^2)=y^{n}$ for large values of $n$ resembles with that of circle $x^2+y^2=1$. This resemblance increases Further as the value of $n$ increases.

Another Fascinating thing I observed is that The Graph of
$\ln (ax^2+by^2)=y^{n}$ for large values of $n$ resembles with the graph of ellipse $ax^2+by^2=1$. So our circle is the special case of the second case, the ellipse.

Can we offer any satisfactory explanation for this?

REPLY [4 votes]: For $-1\leq y\leq 1$, if $n$ is very large, then $\ln(x^2+y^2)\to 0$, so that $x^2+y^2\to 1$. For $y>1$ or $y<-1$, $x$ is very large for large $n$, hence you weren't able to see those points on the graph. The explanation for the ellipse is of the same nature.
An interesting point to note for the ellipse case would be, if the coefficient of $y^2$ in the equation is less than $1$, then only a part of the ellipse would be obtained. This is because in this case, for an ellipse there would be points where $|y|>1$, but such points cannot be seen on the plot due to the same reason I explained in the previous paragraph.