# Use implicit differentiation to evaluateConsider 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...

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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### 1 Answer

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.**