+0

0
1440
6

Jan 9, 2015

#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  :)

Jan 9, 2015

#1
0

it = 4 7

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 !!!!

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P.S. - I found an internet source that says the strange graph is known as a "degenerate" conic....!!!!   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 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....!!!!   Jan 9, 2015
#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
#6
0

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

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

2x^2 -6xy +1xy -3y^2 =0

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

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

x-3y = 0 or 2x+y =0

x =3y or 2x =-y

x= -y/2

Jan 9, 2015