How do you convert $x^{2} + y^{2} = 4 x + 4 y$ $x^2 + y^2 = 4x + 4y$ to polar form?

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How do you convert $x^{2} + y^{2} = 4 x + 4 y$ $x^2 + y^2 = 4x + 4y$ to polar form?

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Answer 1 · 2016-10-15T20:10:12

$r = 4 cos θ + 4 sin θ$ $r=4costheta+4sintheta$

Explanation:

$x = r cos θ$ $x=rcostheta$
$y = r sin θ$ $y=rsintheta$
$L H S = x^{2} + y^{2} = r^{2} {cos}^{2} θ + r^{2} {sin}^{2} θ = r^{2} ({cos}^{2} θ + {sin}^{2} θ) = r^{2}$ $LHS=x^2+y^2=r^2cos^2theta+r^2sin^2theta=r^2(cos^2theta+sin^2theta)=r^2$
As ${cos}^{2} θ + {sin}^{2} θ = 1$ $cos^2theta+sin^2theta=1$
$R H S = 4 r cos θ + 4 r sin θ = 4 r (cos θ + sin θ)$ $RHS=4rcostheta+4rsintheta=4r(cos theta+sintheta)$
LHS=RHS
$r^{2} = 4 r (cos θ + sin θ)$ $r^2=4r(costheta+sintheta)$
Finally $r = 4 (cos θ + sin θ)$ $r=4(costheta+sintheta)$

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How do you convert $x^{2} + y^{2} = 4 x + 4 y$ $x^2 + y^2 = 4x + 4y$ to polar form?

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