I just want to know the name of this equation below :
$$4y^4-45y^2+36y-9x^2=0$$
Actually i was obtanied from this following :
$$|z+2i|+|z-2i|=6$$
When i solve the locus representation of that it gives me a strange graph. Then i checked the solution manual and it said that the graph is ellipse.
Here is my progress :
$\begin{align} &|x+i(y+2)|=6-|x+i(y-2)|\\ &\Leftrightarrow x^2 +(y+2)^2=36-12\sqrt{x^2+(y-2)^2} +(x^2+(y-2)^2)\\ &\Leftrightarrow 8y^2-36=-12\sqrt{x^2+(y-2)^2}\\ &\Leftrightarrow 4y^4-36y^2+36=9(x^2+(y-2)^2)\\ &\Leftrightarrow 4y^4-45y^2+36y-9x^2=0 \end{align}$
Where is my mistake?
My graph is ellipse, but has hyperbola on the outside
But... If i don't simplify this, i mean i just input this equation below (from my work at second line) in GeoGebra and it gives me the actual ellipse :
$$(y+2)^2-(y-2)^2-36=-12\sqrt{x^2+(y-2)^2}$$