The graph of the equation $x^{2}+xy+y^{2} = 2$ is an ellipse lying obliquely in the plane, as illustrated in the figure below.

a.  Compute $\displaystyle \frac{dy}{dx}$.

$\displaystyle \frac{dy}{dx} =$ .

b.  The ellipse has two horizontal tangents. Find an equation of the lower one.

The lower horizontal tangent line is defined by the equation $y =$ .

c.  The ellipse has two vertical tangents. Find an equation of the rightmost one.

The rightmost vertical tangent line is defined by the equation $x =$ .

d.  Find the point at which the rightmost vertical tangent line touches the ellipse.

The rightmost vertical tangent line touches the ellipse at the point .

Hint: The horizontal tangent is of course characterized by $\frac{dy}{dx} = 0$. To find the vertical tangent use symmetry, or solve $\frac{dx}{dy} = 0$.