For how many values of the real number alpha do the two curves y = alpha*x² + alpha*x + 1/24 and x = alpha*y² + alpha*y + 1/24 have a common tangent line?

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2. Since the two curves are symmetric about y = x, any common tangent must be reflected to itself, implying slope = 1 or slope = -1. Solving the tangency conditions for each slope yields a total of 2 values of alpha.

For how many values of the real number alpha do the two curves y = alphax² + alphax + 1/24 and x = alphay² + alphay + 1/24 have a common tangent line?

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