#5**+10 **

Thank you Chris for discovering this for us

This Wikipaedia site that Chris has referred us too is interesting.

http://www.digplanet.com/wiki/Degenerate_conic

Unlike Chris I like to put my feet up and watch video clips. There are tabs at the the top and one is for youtube clips.

I just watched the first one and really liked it.

--------------------------------------------

Actually I just found another simple page one conics.

My knowledge of conics was worse than Chris's in the first place so this is quite enlightening.

http://www.sparknotes.com/math/precalc/conicsections/section1.rhtml

This is what I have learned:

The general form of a conic is

$$Ax^2+Bxy+Cy^2+Dx+Ey+F=0\\

Now if B=0 we have\\

Ax^2+Cy^2+Dx+Ey+F=0\\

If A=C it is a circle\\

If A\ne C \;\; \mbox{BUT A and C have the same sign then it is an ellipse}\\

If A\ne C \;\; \mbox{AND A and C have different signs then it is an hyperbola}\\$$

---------------------------------------------------------

**Ours has an xy term so B is not equal to zero - I do not know what that makes it.**

It is a degenerate conic that is for sure but i am not sure which one.

I would really like more imput from other mathematicians :)

Melody
Jan 9, 2015

#2**+10 **

2x^2-3y^2=5xy and-3x+y=5 rearranging the second equation, we have y = 3x + 5

And putting this into the first equation, we have

2x^2 - 3(3x + 5)^2 = 5x(3x + 5) simplify

2x^2 - 3(9x^2 + 30x +25) = 15x^2 + 25x

2x^2 -27x^2 - 90x - 75 = 15x^2 + 25x

-13x^2 - 27x^2 - 115x - 75 = 0 multiply through by -1

13x^2 + 27x^2 + 115x + 75 = 0

40x^2 + 115x + 75 = 0 divide through by 5

8x^2 + 23x + 15 = 0 factor

(x + 1 )(8x + 15) = 0

And setting each factor to 0, we have that x = -1 and x = -15/8

And using y = 3x + 5 when x = -1, y = 2 and when x = -15/8, y = -5/8

So....our solutions are (-1, 2) and (-15/8, -5/8)

Here's a graph.......https://www.desmos.com/calculator/mdhdt8ytqq

The "blue" function is a straight line.....the other one is something I haven't seen before....a pair of intersecting lines that are "rotated"....very odd !!!!

--------------------------------------------------------------------------------------------------------------

P.S. - I found an internet source that says the strange graph is known as a "degenerate" conic....!!!!

CPhill
Jan 9, 2015

#3**+5 **

This is the most unexpected graph ever. Mmm one for the interest posts definitely !

Thanks for bringing it to my attention Chris

Melody
Jan 9, 2015

#4**+10 **

A little more on this strange graph.....

Apparently, if this can be factored into this form....(x + y) (x - y) = 0, we have a graph of intersecting lines...let's see...

2x^2 - 3y^2 = 5xy

2x^2 - 5xy - 3y^2 = 0

(2x + y) (x - 3y) = 0

Since this "reducible" to this form, this is a degenerate conic that will form two intersecting lines.

Here's the graph, again.....https://www.desmos.com/calculator/hwmmfcew1u

Notice something......if we set the first term in the above factorization to 0, we have 2x + y = 0, or just y = -2x...and this is the line ine on the graph that "falls" from right to left!!! Similarly, doing the same thing to the second factored term produces y = (1/3)x.....and this is the other line on the graph that "rises" from left to right....!!!

And notice one last thing......just like we might do in a quadratic by "factoring and setting to 0" to find the roots....we are doing something similar here....except that, instead of generating "roots," we're generating * equations* of lines....!!!!

Here's some more info about these odd graphs....http://www.digplanet.com/wiki/Degenerate_conic

CPhill
Jan 9, 2015

#5**+10 **

Best Answer

Thank you Chris for discovering this for us

This Wikipaedia site that Chris has referred us too is interesting.

http://www.digplanet.com/wiki/Degenerate_conic

Unlike Chris I like to put my feet up and watch video clips. There are tabs at the the top and one is for youtube clips.

I just watched the first one and really liked it.

--------------------------------------------

Actually I just found another simple page one conics.

My knowledge of conics was worse than Chris's in the first place so this is quite enlightening.

http://www.sparknotes.com/math/precalc/conicsections/section1.rhtml

This is what I have learned:

The general form of a conic is

$$Ax^2+Bxy+Cy^2+Dx+Ey+F=0\\

Now if B=0 we have\\

Ax^2+Cy^2+Dx+Ey+F=0\\

If A=C it is a circle\\

If A\ne C \;\; \mbox{BUT A and C have the same sign then it is an ellipse}\\

If A\ne C \;\; \mbox{AND A and C have different signs then it is an hyperbola}\\$$

---------------------------------------------------------

**Ours has an xy term so B is not equal to zero - I do not know what that makes it.**

It is a degenerate conic that is for sure but i am not sure which one.

I would really like more imput from other mathematicians :)

Melody
Jan 9, 2015