graph{(x^2+y^2+y)sqrt(x^2+y^2)=4x^2 [-8.046, 8.046, -4.02, 4.024]}
From the given
r=-sin theta+4*cos^2 thetar=−sinθ+4⋅cos2θ
the conversion equations:
r=sqrt(x^2+y^2)r=√x2+y2 and theta=tan^-1(y/x)θ=tan−1(yx)
tan theta=y/xtanθ=yx and sin theta=y/sqrt(x^2+y^2)sinθ=y√x2+y2 and cos theta=x/sqrt(x^2+y^2)cosθ=x√x2+y2
Convert now, the given equation becomes
sqrt(x^2+y^2)=-y/sqrt(x^2+y^2)+(4*x^2)/(x^2+y^2)√x2+y2=−y√x2+y2+4⋅x2x2+y2
multiply both sides of the equation by (x^2+y^2)(x2+y2)
(x^2+y^2)sqrt(x^2+y^2)=-(y*(x^2+y^2))/sqrt(x^2+y^2)+((x^2+y^2)(4*x^2))/(x^2+y^2)(x2+y2)√x2+y2=−y⋅(x2+y2)√x2+y2+(x2+y2)(4⋅x2)x2+y2
(x^2+y^2)sqrt(x^2+y^2)=-(ycancel((x^2+y^2)))/cancelsqrt(x^2+y^2)+(cancel((x^2+y^2))(4*x^2))/cancel(x^2+y^2)
(x^2+y^2)sqrt(x^2+y^2)=-y*sqrt(x^2+y^2)+4*x^2
(x^2+y^2)sqrt(x^2+y^2)+y*sqrt(x^2+y^2)=4*x^2
(x^2+y^2+y)*sqrt(x^2+y^2)=4*x^2
