Problem

An interesting curve first studied by Nicomedes around 200 B.C. is the conchoid, which has the equation

Use implicit differentiation to find a tangent line to this curve at the point (–1, 0).

Solution

Differentiating both sides of the equation with respect to x, we get

Solve for dy/dx.

Does this give us the slope at ( –1, 0)? Substitute these values for x and y.

It appears from the graph that there are actually two tangent lines at this point, since the curve intersects itself, which may be why dy/dx cannot help us, since it would only produce one value. Let's go back to the original equation for the curve to see what we can figure out. The curve is not a function, but can be written as the union of two functions, similar to the way a circle can be thought of as two functions joined together:

For the conchoid, solving for y in terms of x produces two functions of x:

We can compute dy/dx for the red and blue components of the curve separately.

Since blue(x) is the negative of red(x), we get

These are the slopes of the tangent lines. The equations of the tangent lines are