How do we write the polar equation $r = 3 tan θ$ $r=3tantheta$ in Cartesian coordinates?

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Answer 1 · 2017-07-18T08:53:41

$9 y^{2} = x^{4} + x^{2} y^{2}$ $9y^2=x^4+x^2y^2$

we need the conversion formulae

$r^{2} = x^{2} + y^{2} - - (1)$ $r^2=x^2+y^2--(1)$

$x = r cos θ - - - (2)$ $x=rcostheta---(2)$

$y = r sin θ - - - (3)$ $y=rsintheta---(3)$

we have

$r = 3 tan θ$ $color(red)(r=3tantheta)$

we have $tan θ$ $tantheta$ so $(3) \div (2)$ $(3)-:(2)$

$y/x=(cancel(r)sintheta)/(cancel(r)costheta)=tantheta---(4)$

from $(1)" "$ we have $" "r=sqrt(x^2+y^2)---(5)$

$(4)" "(5)" "$ into the original equation

$color(red)(sqrt(x^2+y^2)=3y/x$

square both sides

$x^2+y^2=(9y^2)/x^2$

$:.9y^2=x^4+x^2y^2$

How do we write the polar equation r=3tanthetar=3tanθr=3tantheta in Cartesian coordinates?