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Tangent Curves

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Y=alpha*x^2 + alpha*x + (1/24)

x=alpha*y^2 + alpha*y + (1/24)

These are two curves.
For what values of alpha are these two curves tangent to each other.

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The two curves are
For , we have
For , we have ...

Solution Summary

Values are found to make two curves tangent to each other. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.