Parameterize Tangent Line to Level Curve

Question

$$f(x, y) = x^2y + yx - xy^2$$

Find a parameterization of the tangent line to the level curve of $f$ through the point $(1, 1)$.

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I have already computed the gradient of $f$ and found the critical points.

Any help would be appreciated. Thanks!

Answer 1

You have that $\left(\frac{\partial f}{\partial x}(1, 1), \frac{\partial f}{\partial y}(1, 1) \right)$ is normal to the level curve. Then the vector $\left(-\frac{\partial f}{\partial y}(1, 1), \frac{\partial f}{\partial x}(1,1) \right)$ is tangent to the level curve. I just rotated the gradient by $90 $ degrees counterclockwise. Then your line is: $${\bf X}(t) = (1,1)+ t \left(-\frac{\partial f}{\partial y}(1, 1), \frac{\partial f}{\partial x}(1,1) \right), \quad t\in \Bbb R.$$