# Tangent line to level curve

Introduction: When we have to determine the tangent line to a level curve, then we can start by solving the equation for the level curve for $y$ and use that to determine the tangent line to the graph of the function $y(x)$. We have done this in Example 4 at level curves. Unfortunately, sometimes it is not possible to rewrite the level curve. However, even in that case we can determine the slope of the tangent line to the level curve by the use of the following rule.

Theorem: The slope of the tangent line to the level curve of the function $z(x,y)$ at the point $(x,y)$ of the level curve is given by
$$\text{rc} = -\dfrac{z'_x(x,y)}{z'_y(x,y)}.$$