how to graph these curves?

[KataKoniK]

Hi,

I was wondering how would I graph a sketch of these curves without knowing any values of them? They are orthogonal trajectories btw


i) x^2 + y^2 = ax
x^2 + y^2 = by

ii) y = ax^3
x^2 + 3y^2 = b

Thanks in advnace.

[HallsofIvy]

x2+ y2= ax is the same as

x2- ax+ y2= 0. Now complete the square in the x variable:

x2- ax+ a2/4+ y2= a2/4
(x- a/2)2+ y2= a2/2.

That's a circle with center at (a/2, 0) and radius a/2 (so it goes through (0,0)).

x2 + y2 = by

Same thing: a circle with center at (0, b/2) and radius b/2 (and goes through (0,0)).

y = ax3
Well, that's just a pretty standard cubic. Goes through(-2,-8a), (-1, -a), (0,0), (1, a), (2, 8a), etc.

x2 + 3y2 = b
Same as x2/b + y2/(b/3)= 1,
an ellipse with center at (0,0) vertices (√(b), 0), (-&radic(b),0), (0, √(b/3)), (0,-√(b/3)).

[KataKoniK]

Thanks a lot man. Really appreciate it.