Start with the given
3xy=-x^2-2y^23xy=−x2−2y2
x^2+3xy+2y^2=0x2+3xy+2y2=0
do factoring to simplify
(x+y)(x+2y)=0(x+y)(x+2y)=0
Use x=r cos thetax=rcosθ and y=r sin thetay=rsinθ
(x+y)(x+2y)=0(x+y)(x+2y)=0
(r cos theta+r sin theta)(r cos theta+2*r sin theta)=0(rcosθ+rsinθ)(rcosθ+2⋅rsinθ)=0
cancel all the rrs
(cos theta+ sin theta)( cos theta+2* sin theta)=0(cosθ+sinθ)(cosθ+2⋅sinθ)=0
equate both factors to zero
cos theta+ sin theta=0cosθ+sinθ=0
sin theta=-cos thetasinθ=−cosθ
tan theta=-1tanθ=−1
theta=(3pi)/4=135^@θ=3π4=135∘
For the other factor:
cos theta+2* sin theta=0cosθ+2⋅sinθ=0
2*sin theta=-cos theta2⋅sinθ=−cosθ
tan theta=-1/2tanθ=−12
theta=tan^-1 (-1/2)=153.435^@θ=tan−1(−12)=153.435∘
These are 2 lines passing thru the Origin (0, 0) with
slopes =-1 and -1/2
graph{3xy=-x^2-2y^2[-20,20,-10,10]}
have a nice day !
