find the points on the curve x^2+xy+y^2=7 (a) where the tangent is parallel to the x-axis and (b) where the tangent is parallel to the y-axis. In the latter case, dy⁄dx is not defined, but dx⁄dy is. What value does dx⁄dy have at these points?

Answer 1

to find dy/dx,

2x + y + xy' + 2yy' = 0
y' = -(2x+y)/(x+2y)
y'=0 where 2x+y=0

x^2 + x(-2x) + (-2x)^2 = 7
x^2 - 2x^2 + 4x^2 = 7
3x^2 = 7
x = √(7/3)
y = -2√*7/3)

interchange the variables for dx/dy

naturally, when dy/dx=0, dx/dy is undefined, and vice-versa.

To see this all laid out, visit

wolframalpha.com and enter

plot x^2+xy+y^2=7, y^2 = 28/3, x^2 = 28/3