What does this equation represent ?

[Qwertyuiop[]]

What does the equation x^2 + y^2 +2 = 0 define in Cartesian plane ? I've never seen this equation before and it's not in the form y=mx +b so it's not a straight line equation. Is it b) pair of parallel lines (I didn't know parallel lines had an equation ??) c) point (point has an equation ? ??) d) empty set has an equation too ?
or d) pair of incident lines (and what is this ? LOL)

 

[Qwertyuiop[]]

I tried graphing this eqn by testing 3 negative and 3 positive x-value . In each case I get something like this y^2 = -a . I can not take the square root of negative number so this must be an empty set.

 

[Deleted member 4993]

What does the equation x^2 + y^2 +2 = 0 define in Cartesian plane ? I've never seen this equation before and it's not in the form y=mx +b so it's not a straight line equation. Is it b) pair of parallel lines (I didn't know parallel lines had an equation ??) c) point (point has an equation ? ??) d) empty set has an equation too ?
or d) pair of incident lines (and what is this ? LOL)

Hint:

x^2 + y^2 ≥ 0 in real domain. So

x^2 + y^2 + 2 = 0 does NOT have a solution in the real domain. We cannot represent that expression with a graph in real domain.

 

[Cubist]

I didn't know parallel lines had an equation ??

Here's an example
[math](x−y−1)^2 (x−y+1)^2 = 0[/math]can you see how it works?

c) point (point has an equation ?

How many (x,y) coordinates would satisfy the following equation?
[math]x^2+y^2 = 0[/math]

d) pair of incident lines (and what is this ? LOL)

It's a fancy way of saying two lines that cross each other

 

[Dr.Peterson]

What does the equation x^2 + y^2 +2 = 0 define in Cartesian plane ? I've never seen this equation before and it's not in the form y=mx +b so it's not a straight line equation. Is it b) pair of parallel lines (I didn't know parallel lines had an equation ??) c) point (point has an equation ? ??) d) empty set has an equation too ?
or d) pair of incident lines (and what is this ? LOL)

This question is about "degenerate conic sections" (or "degenerate conics"), which can take any of the forms listed. You may want to look that up; see, for example, here:

Degenerate conic - Wikipedia

en.wikipedia.org

https://math.sierracollege.edu/math_faculty/jduran/Trig%20Handouts/ConicOverview.pdf

8.5: Rotation of Axes

In this section, we will learn how to define any conic in the polar coordinate system in terms of a fixed point, the focus at the pole, and a line, the directrix, which is perpendicular to the polar …

math.libretexts.org

Start with the second or third, which have pictures and don't go quite as deep.

 

[Qwertyuiop[]]

This question is about "degenerate conic sections" (or "degenerate conics"), which can take any of the forms listed. You may want to look that up; see, for example, here:

Degenerate conic - Wikipedia

en.wikipedia.org

https://math.sierracollege.edu/math_faculty/jduran/Trig%20Handouts/ConicOverview.pdf

8.5: Rotation of Axes

In this section, we will learn how to define any conic in the polar coordinate system in terms of a fixed point, the focus at the pole, and a line, the directrix, which is perpendicular to the polar …

math.libretexts.org

Start with the second or third, which have pictures and don't go quite as deep.

This is exactly what i was looking for! thanks. equations of all the options given in the question.

 

[Qwertyuiop[]]

Here's an example
[math](x−y−1)^2 (x−y+1)^2 = 0[/math]can you see how it works?


How many (x,y) coordinates would satisfy the following equation?
[math]x^2+y^2 = 0[/math]

It's a fancy way of saying two lines that cross each other

Hmm so parallel lines equation it's just the product of two linear equations. That makes sense.

"How many (x,y) coordinates would satisfy the following equation?"- only (0 , 0). So it's a point at the origin!

 

[Steven G]

Suppose we have the lines 3x+2y=11 and 5x - 4y = 12. These two lines are NOT parallel.

Now 3x+2y=11 is the same as 3x+2y-11=0 and 5x-4y =12 is the same as 5x-4y-12=0.

Consider their product (3x+2y-11)(5x-4y-12) = 0

When does a product equal zero? When one or more of the factors equal 0.

So when does 3x+2y-11=0? Exactly when 3x+2y =11
When does 5x-4y-12=0? Exactly when 5x-4y=12

The graph of this product (3x+2y-11)(5x-4y-12) = 0 is the two lines from the very top line. And they are NOT parallel.
Personally I never thought of an equation that has two lines before.

 

[Cubist]

Hmm so parallel lines equation it's just the product of two linear equations. That makes sense.

"How many (x,y) coordinates would satisfy the following equation?"- only (0 , 0). So it's a point at the origin!

Indeed, well done!

I didn't actually need to square anything, since the following also describes two parallel lines...
[math](x−y−1) (x−y+1) = 0[/math]...therefore I'm learning too

 

[Steven G]

Here's an example
[math](x−y−1)^2 (x−y+1)^2 = 0[/math]can you see how it works?

Hmm so parallel lines equation it's just the product of two linear equations. That makes sense.

"How many (x,y) coordinates would satisfy the following equation?"- only (0 , 0). So it's a point at the origin!

Let's see if (0,0) is a solution.
(x-y-1)^2(x-y+1)^2=0
(0-0-1)(0-0+1) = (-1)(1)=-1 ≠ 0.
So (0,0) is not a solution.

 

[Cubist]

Let's see if (0,0) is a solution.
(x-y-1)^2(x-y+1)^2=0
(0-0-1)(0-0+1) = (-1)(1)=-1 ≠ 0.
So (0,0) is not a solution.

I think they were responding to this question...

How many (x,y) coordinates would satisfy the following equation?
[math]x^2+y^2 = 0[/math]