Consider the curve defined by x^2+xy +y^2=27
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.
The Attempt at a Solution
a. I used implicit differentiation
x^2 +xy +y^2 =27
2x +y + xy' +2yy' = 0
y'(x+2y)= - (2x+y)
y' = -(2x+y)/(x+2y). Im pretty sure this is right.
b. i know that the lines must have the same slope for them to be parallel, but im not actually sure what else to do with this. how do u find the x-intercepts and show that their slopes are equal?
c. the vertical line would be like x=k, then you plug k into the orginal curve and then solve for k? i got 6 and -6 is that right? can someone please help me with this