Translate \(\displaystyle{2}{x}{y}={x}^{{2}}-{y}^{{2}}\) to polar coordinates Why it confuses

Rosa Townsend

Rosa Townsend

Answered question

2022-04-10

Translate 2xy=x2y2 to polar coordinates
Why it confuses me:
x=rcosϕ
y=rsinϕ
Then:
2r2cos{ϕ}sin{ϕ}=r2cos{ϕ}r2sin{ϕ}=r2(cos2{ϕ}sin2{ϕ})
r2sin{2ϕ}=r2cos{2ϕ}
sin{2ϕ}=cos{2ϕ}
Which gives:
ϕ=π8+πn2,     nZ
But it's just a set of ϕ values. How do i plot such a function? In polar coordinates r is a function of some ϕ, isnt that?

Answer & Explanation

szalbierzfytg

szalbierzfytg

Beginner2022-04-11Added 13 answers

After conversion to polar coordinates, r disappears from the equations (but for r=0, which corresponds to the origin). This means that for a given ϕ,r is arbitrary and this describes an infinite straight line through the origin and with direction ϕ.
By the way, you can obtain the same result by plugging the equation of a straight line through the origin, y=mx, which gives
2mx2=x2m2x2
or
2m=1m2
or
m=±21
riasc31lj

riasc31lj

Beginner2022-04-12Added 8 answers

It is
ϕ=π8+πn2,    nZ
and not x.
So
x=rcos(π8+πn2)=
=r(cosπ8cosπn2sinπ8sinπn2)
Now try to put n=0,1,2,3,4.... We see that:
if we put n=0,4,8... we get the same value. if we put n=1,5,9... we get the same value. ...
So write n=4k+r where r{0,1,2,3} and you will get 4 different families of solution.

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