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## Homework Statement

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