Bézout's Theorem/Examples/Parabola and Tangent Line

Example of Use of Bézout's Theorem

A parabola $\PP$ and a tangent to $\PP$ meet at a point of multiplicity $2$. Two parallel straight lines still meet at one point at infinity.


Proof

A straight line is of degree $1$.

A parabola is of degree $2$.

Hence, by Bézout's Theorem, their point of intersection is of multiplicity $2 \times 1 = 2$.

$\blacksquare$


Sources

  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Bézout's theorem
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