How can I determine wheter or not two curves are tangents?

calculus multivariable-calculus curves tangent-line

Solution 1:

Solving simultaneously leads to the quadratic equation $$x^2(a^2+b^2)-2a^2x+a^2-a^2b^2=0$$

The discriminant can be simplified as $$4a^2b^2(a^2+b^2-1)$$

Tangency means double roots, and this is zero since $a^2+b^2=1$

Hence the result.

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