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August 6th, 2012, 10:57 AM   #1
ehh
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rotation of axes question

x^2 + 4xy + y^2 - 3 = 0

rewrite the equation in a rotated x'y' system without ans x'y' term. express the equation involving x' and y' in the standard form of a conic section.

my solution is x'^2 - y'^2/3 = 1

would anyone like to check this for me ?
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August 6th, 2012, 01:16 PM   #2
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Re: rotation of axes question

It would help if you would supply the equations transforming (x,y) to (x',y').
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August 6th, 2012, 01:35 PM   #3
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Re: rotation of axes question

first you want to find the angle you are rotating, using the formula 1/2(Tan^(-1))(B/(A-C)) is A does not equal to C, however, in this case, they do equal to each other, meaning the angle of the rotation is 45 degrees.
let ? = the angle of rotation
A'=Acos^2(?)+Bsin?cos?+Csin^2(?)
B'=0
C'=Asin^2(?)-Bsin?cos?+Ccos^2(?)
D'=0
E'=0
F'=-3

D' and E' are both = 0 because D and E are 0
so A'=1/2+4x(1/2)+1/2
C'=1/2-4x(1/2)+1/2
F'=-3

solve each equation
A'=3
C'=-1
F'=-3

Finally, the equation i have is:
3x'^2-y'^2-3=0
if we divide everything by 3 (i believe that is what you did)
x'^2-(y'^2)/3-1=0, Which is exactly what you got, i believe you are correct
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August 6th, 2012, 02:22 PM   #4
ehh
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Re: rotation of axes question

Quote:
Originally Posted by mathman
It would help if you would supply the equations transforming (x,y) to (x',y').
huh? i gave you the problem as it is writen.

Quote:
Originally Posted by wuzhe
first you want to find the angle you are rotating, using the formula 1/2(Tan^(-1))(B/(A-C)) is A does not equal to C, however, in this case, they do equal to each other, meaning the angle of the rotation is 45 degrees.
let ? = the angle of rotation
A'=Acos^2(?)+Bsin?cos?+Csin^2(?)
B'=0
C'=Asin^2(?)-Bsin?cos?+Ccos^2(?)
D'=0
E'=0
F'=-3

D' and E' are both = 0 because D and E are 0
so A'=1/2+4x(1/2)+1/2
C'=1/2-4x(1/2)+1/2
F'=-3

solve each equation
A'=3
C'=-1
F'=-3

Finally, the equation i have is:
3x'^2-y'^2-3=0
if we divide everything by 3 (i believe that is what you did)
x'^2-(y'^2)/3-1=0, Which is exactly what you got, i believe you are correct
thank you! i wouldve typed out my work but its so much haha.
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