Consider the equation x^2 + xy + y^2 = 1

Question

Find all points where the tangent line is parallel to the line y = −x.

Thank you for helping

Answer 1

well does that not mean that dy/dx = -1?

Using implicit differentiation,
2x + y + xy' + 2yy' = 0
y' = -(2x+y)/(x+2y)

so, you want

(2x+y)/(x+2y) = 1
2x+y = x+2y
y = x

plug that into the original equation and you get

x^2 + x^2 + x^2 = 1
x^2 = 1/3

so, at (1/√3,1/√3) and (-1/√3,-1/√3) dy/dx = -1

See from the URL below that this is true.

https://www.wolframalpha.com/input/?i=x^2+%2B+xy+%2B+y^2+%3D+1

Answer 2

Thank youuu so much