Question 22

A contour of the magnetic potential $U\,$ in a specialized laboratory setup is modeled by the curve with equation

x^5 y + 4x y^5 = 130

Prove that the curve does not intersect the coordinate axes.

Show that

\frac{dy}{dx} = -\frac{5x^4 y + 4y^5}{x^5 + 20xy^4}

Prove that the curve has no stationary points.

In the case when $x > 0$ , find the equation of the tangent line to the curve at the point where $y = 2$ . Give your answer in the form $a y + b x = c$ , where $c\,$ are integers.