Use implicit differentiation to evaluate
Consider the curve defined by x^2+xy+y^2=27 Write an expression for the slope of the curve at any point (x,y) and determine whether the lines tangent to the curve at the x-intercepts of the curve are parralel.
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You need to differentiate the function wth respect to x such that:
`2x + (x'*y + x*y') + 2y*y' = 0`
`2x + y + x*y' + 2y*y' = 0`
You need to isolate terms containing y' to the left side such that:
`x*y' + 2y*y' = -2x - y`
You need to factor out y' such that:
`y'(x + 2y) =-2x - y =gt y' = (-2x - y)/(x + 2y)`
You need to remember that curve intercepts x axis at y=0, hence, substituting 0 for y in equation of curve yields:
`y^2 =27 =gt y = +-3sqrt3`
You need to substitute 0 for y in equation of derivative such that:
`y' = -(2x)/x =gt y' = -2`
Hence, this proves that tangent line at x intercepts are parallel since the slope `y'=m=-2` is the same for both tangent lines.
Hence, the equation of the slope of the curve at any point is `y' = (-2x - y)/(x + 2y)` and the since the slopes of tangent lines at x intercepts have equal values, the tangent lines are parallel.
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