(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the points (if any) of of horizontal tangent lines on :

x^{2}+ xy + y^{2}= 6

2. Relevant equations

n/a

3. The attempt at a solution

So far I've concluded that I must find the points at which dy/dx = 0. I've solved for dy/dx and arrived at dy/dx = (-2x-y)/(x+2y)

I assume that I would just have to get a "0" in the numerator to satisfy the horizontal tangent but doing so gives me

-2x-y = 0 ==> y = -2x

This seems that there would be an infinite number of horizontal tangents (as long as the original denominator didn't equal "0") but the graph of the original equation, per Wolfram Alpha, seems to be an ellipse so I'm only looking for two solutions...

Have I missed a component of the concept or should I not be ending up with an ellipse?

Thank you

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# Horizontal tangents via implicit differentiation

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