# Derivatives and parallel lines

## 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

Shooting Star
Homework Helper
a. Correct

b. Find x-intercepts of the curve by putting y=0, and find the x values. Then put these x and y in dy/dx.

c. Find where dx/dy is zero.

Office_Shredder
Staff Emeritus
Gold Member
(a) is almost right, but you neglect the possibility that x+2y=0. You would either have to find an alternative expression for that case, or show it never happens on the curve (e.g. by substituting x=-2y into the original equation)

For (b), I suggest being really slick and not calculating where the x-intercepts are located if you can (try plugging y=0 into your dy/dx expression. Where can this go wrong?). Note a line that's tangent to a curve has a slope equal to dy/dx

For (c), if the curve is parallel to a vertical line, then dy/dx is +/- infinity. Taking a page from part (a), when does this occur?

Shooting Star
Homework Helper
For (b), I suggest being really slick and not calculating where the x-intercepts are located if you can (try plugging y=0 into your dy/dx expression. Where can this go wrong?). Note a line that's tangent to a curve has a slope equal to dy/dx

I didn't quite get what exactly you meant.

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HallsofIvy
Homework Helper
He is saying, rather than set y= 0 in the original equation, to find the x-intercepts, set y= 0 in the formula for the derivative. It happens, for this particular problem, that you don't need to know x!