Finding a tangent line parallel to the x-axis with dy/dx

Question

1. $x^2+xy+y^2=7$
2. Find $dy/dx$
3. $dy/dx= (-2x-y)/(x+2y)$

How do I take $dy/dx$ and get the equation of the tangent line parallel to the $x$-axis?

Answer 1

You have

$$ x^2 + xy + y^2 = 7. $$

Using implicit differentiation you get $$ \begin{align} \frac{d}{dx} x^2 + xy + y^2 &= \frac{d}{dx} 7 &\Rightarrow\\ 2x + y + x\frac{dy}{dx} + 2y\frac{dy}{dx} &= 0. \end{align} $$

From this you solve for $\frac{dy}{dx}$. You get [you can do this] $$ \frac{dy}{dx} = \frac{-2x-y}{x + 2y} $$ To find the equation of the tangent line that is parallel to the $x$-axis, you need to determine the $x$ and $y$ such that the numerator is zero and the denominator is not zero. At those points you have a slope of zero. Then you can simply write down the equaltion of the tangent line (the slope is obviously zero, so...).