Identifying similar families of cuvers

[shayaan_musta]

Homework Statement

State which of the following families of curves are similar sets.

Homework Equations

1)Y^{2}=4ax
2)Y=acosh(\frac{x}{a})
3)\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1
4)Y=2a^{3}log\frac{x}{a^{3}}
5)btan^{-1}\frac{y}{x}=a+y
6)x^{3}+y^{3}=3axy

The Attempt at a Solution

1)parabola
2)Parabola
3)hyperbola
4)I don't know which type of this curve is!
5)I don't know which type of this curve is!
6)this seems like circle. but in circle both terms x and y are squared and here is cube.

Kindly tell me whether I am wrong in guessing families or not??
Thanks

 

[HallsofIvy]

I am not sure what you mean by "similar" but only the first and third are conic sections- and the third is an ellipse, not a hyperbola. (2 "looks like" a parabola but is not.)

 

[shayaan_musta]

I am not sure what you mean by "similar" but only the first and third are conic sections- and the third is an ellipse, not a hyperbola. (2 "looks like" a parabola but is not.)

Similar means like families.
Ok yes 3rd one is ellipse. But what about 2,4,5 and 6. How to deduce the types of these curves? Any hint will be appreciable.