The graph of the equation [math] is an ellipse lying obliquely in the plane, as illustrated in the figure below.

a.  Compute [math].

[math] .


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

The lower horizontal tangent line is defined by the equation [math] .


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

The rightmost vertical tangent line is defined by the equation [math] .


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 [math]. To find the vertical tangent use symmetry, or solve [math].