Classification of curves and surfaces linear algebra
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I have been given the following question on a homework sheet:
I was wondering if I could have some help. I have this so far:
x2−y2+2xy−1=(x+y)2−2y2−1=0x2−y2+2xy−1=(x+y)2−2y2−1=0
and so (x+y)2−2y2=1(x+y)2−2y2=1.
I can clearly see that this is a hyperbola. But if I let
x′=x+yx′=x+y and
y′=yy′=y
so that it is in the correct form, I believe this actually distorts the curve.
How can I make a suitable change of co-ordinates to show that it is a hyperbola without distorting the curve? My thoughts are along the same lines as normalisation and orthogonal bases, but I am just looking for some guidance as to whether I am heading in the correct direction and a way to do so without distorting the curve.
I was wondering if I could have some help. I have this so far:
x2−y2+2xy−1=(x+y)2−2y2−1=0x2−y2+2xy−1=(x+y)2−2y2−1=0
and so (x+y)2−2y2=1(x+y)2−2y2=1.
I can clearly see that this is a hyperbola. But if I let
x′=x+yx′=x+y and
y′=yy′=y
so that it is in the correct form, I believe this actually distorts the curve.
How can I make a suitable change of co-ordinates to show that it is a hyperbola without distorting the curve? My thoughts are along the same lines as normalisation and orthogonal bases, but I am just looking for some guidance as to whether I am heading in the correct direction and a way to do so without distorting the curve.
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