Consider the curve defined by x^2 + xy + y^2=27

Question

anonymous · Answer

a.) Write an expression for the slope of the curve at any point (x,y)
b.) Determine whether the lines tangent to the curve at the x-intercepts of the curve are parallel. Show the analysis that leads to your conclusion
c.) Find the points on the curve where the lines tangent to the curve are vertical

anonymous · Answer

a) Find dy/dx via implicit differentiation. Do you know how to do that?

anonymous · Answer

yeah i got -2x-y/(x+2y) is that correct and the answer to part a.)? @SithsAndGiggles

anonymous · Answer

Yes, that's right. Now find the x-intercepts.

anonymous · Answer

how would i find the x-intercepts

anonymous · Answer

set y=0

anonymous · Answer

to the what i got from doing dy/dx 
-2x-y/(x+2y) = 0 ?

anonymous · Answer

Using the original curve,
$$x^2+xy+y^2=27,$$
and setting y = 0 (as @Hoa suggested, since all x-intercepts have the ordered pair (x, 0) for some x), you have
$$x^2+x(0)+0^2=27\
x^2=27$$