Expand the equation. This is not essential, but it seems easier when differentiating implicitly:
#3(x^2+y^2)^2=100(x^2+y^2)#
#3x^4+6x^2y^2+3y^4-100x^2+100y^2=0#
Differentiating implicitly:
#dy/dx(3x^4+6x^2y^2+3y^4-100x^2+100y^2)=#
#12x^3+12x^2y*dy/dx+12xy^2+12y^3*dy/dx-200x+200y*dy/dx=0#
#12x^2y*dy/dx+12y^3*dy/dx+200y*dy/dx=-12x^3-12xy^2+200x#
Factor:
#dy/dx(12x^2y+12y^3+200y)=-12x^3-12xy^2+200x#
#dy/dx=(-12x^3-12xy^2+200x)/(12x^2y+12y^3+200y)#
#dy/dx=(-3x^3-3xy^2+50x)/(3x^2y+3y^3+50y)=-((3x^3+3xy^2-50x))/(3x^2y+3y^3+50y)#
Let #m=-((3x^3+3xy^2-50x))/(3x^2y+3y^3+50y)#
Using point slope form of a line:
#y-(-2)=m(x-(-4))#
#y+2=m(x+4)#
#y=mx+4m-2#
Pugging in values for #x and y# into #bbm# i.e. #(-4,-2)#
#-((3(-4)^3+3(-4)(-2)^2-50(-4)))/(3(-4)^2(-2)+3(-2)^3+50(-2))=-2/11#
#m=-2/11#
Substituting in:
#y=mx+4m-2#
#y=-2/11x+4(-2/11)-2#
#color(blue)(y=-2/11x-30/11)#
GRAPH:
