What does the equation represent?

[zorro]

Homework Statement

What does ax² + by² = 0 represent if
1)a and b are of same sign
2)a and b are of opposite sign.

Homework Equations

The Attempt at a Solution

1) I think it represents a point as radius of circle is 0.
2) No idea

 

[Deneb Cyg]

Try writing down the equations and putting them in the form y=f(x) taking a and b as 1 for simplicity.

1) You're right, it is a point in that it is a circle with radius 0. But it also has roots containing i so I'm not sure what to actually call it...

x²+y²=0
y²=0-x²
y=sqrt(-x²)
The square root of a negative number is imaginary
y=+ix

2) This one is actually easier when you think about it

x²-y²=0
y²=0+x²
y²=x²
y=+x

You should be able to recognize what that would look like in a graph.

 

[hunt_mat]

The first one would represent the point x=y=0. The second one, a line with y-intercept =0.

 

[Mark44]

Another way to look at this is in terms of specific conic sections. If the equation were ax² + by² = 1, this would be:
a) a circle if a > 0, b > 0 and a = b.
b) an ellipse if a > 0, b > 0 and a != b
c) a hyperbola if a and b are opposite in sign

For the equation ax² + by² = 0, you can think of this as
a) a degenerate circle if a > 0, b > 0 and a = b. I.e., a circle of radius zero, centered at the origin. In other words, a single point.
b) a degenerate ellipse if a > 0, b > 0 and a != b. Also a single point.
c) a degenerate hyperbola if a and b are opposite in sign. I.e., a pair of straight lines that intersect at the origin.

 

[zorro]

Thanks everone!