Convert to Polar Form...1

[nycmathguy]

Can you show me how to do 22 and 24?

 

[MathLover1]

22. x^2+4x+y^2+4y=0

The polar coordinates r (the radial coordinate) and θ (the angular coordinate, often called the polar angle) are defined in terms of Cartesian coordinates by
x = r cos θ
y = r sin θ, where r is the radial distance from the origin, and θ is the counterclockwise angle from the x-axis. In terms of x and y, r = sqrt(x^2 + y^2)
θ = |tan^(-1)(y/x).

so, substitute x = r cos θ ad y = r sin θ in given equation

( r cos (θ) )^2+4( r cos( θ) )+(r sin (θ))^2+4(r sin (θ))=0

r^2 cos ^2(θ) +4r cos (θ) +r^2 sin ^2(θ)+4r sin( θ)=0

r^2( cos ^2(θ) +sin ^2(θ))+4r cos θ +4r sin θ=0

r^2*1+4r cos θ +4r sin θ=0

r(r+4cos θ +4sin θ)=0


24. x^2(x^2+y^2)=y^2

(r cos (θ) )^2((r cos (θ) )^2+(r sin θ)^2)=(r sin θ)^2

r^2cos ^2(θ) (r^2cos ^2(θ) +r^2 sin ^2(θ))=r^2 sin ^2
r^2cos ^2(θ) (r^2(cos ^2(θ) +sin ^2(θ)))=r^2 sin ^2(θ)
r^2cos ^2(θ) (r^2*1)=r^2 sin ^2(θ)
r^2cos ^2(θ) r^2=r^2 sin ^2(θ)
r^4cos ^2(θ) =r^2 sin ^2(θ).........simplify, divide by r
r^3cos ^2(θ) = r sin ^2(θ)

 

[nycmathguy]

22. x^2+4x+y^2+4y=0

The polar coordinates r (the radial coordinate) and θ (the angular coordinate, often called the polar angle) are defined in terms of Cartesian coordinates by
x = r cos θ
y = r sin θ, where r is the radial distance from the origin, and θ is the counterclockwise angle from the x-axis. In terms of x and y, r = sqrt(x^2 + y^2)
θ = |tan^(-1)(y/x).

so, substitute x = r cos θ ad y = r sin θ in given equation

( r cos (θ) )^2+4( r cos( θ) )+(r sin (θ))^2+4(r sin (θ))=0

r^2 cos ^2(θ) +4r cos (θ) +r^2 sin ^2(θ)+4r sin( θ)=0

r^2( cos ^2(θ) +sin ^2(θ))+4r cos θ +4r sin θ=0

r^2*1+4r cos θ +4r sin θ=0

r(r+4cos θ +4sin θ)=0


24. x^2(x^2+y^2)=y^2

(r cos (θ) )^2((r cos (θ) )^2+(r sin θ)^2)=(r sin θ)^2

r^2cos ^2(θ) (r^2cos ^2(θ) +r^2 sin ^2(θ))=r^2 sin ^2
r^2cos ^2(θ) (r^2(cos ^2(θ) +sin ^2(θ)))=r^2 sin ^2(θ)
r^2cos ^2(θ) (r^2*1)=r^2 sin ^2(θ)
r^2cos ^2(θ) r^2=r^2 sin ^2(θ)
r^4cos ^2(θ) =r^2 sin ^2(θ).........simplify, divide by r
r^3cos ^2(θ) = r sin ^2(θ)

Thank you.